Research ArticleA New Class of Contraction in 119887-Metric Spaces and Applications

Preeti Kaushik1 Sanjay Kumar1 and Kenan Tas2

1Department of Mathematics DCRUST Murthal Sonepat 131 039 India2Department of Mathematics and Computer Science Cankaya University Ankara Turkey

Correspondence should be addressed to Preeti Kaushik preeti1785gmailcom

Received 7 March 2017 Revised 27 April 2017 Accepted 8 May 2017 Published 11 June 2017

Academic Editor Kunquan Lan

Copyright copy 2017 Preeti Kaushik et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A novel class of 120572-120573-contraction for a pair of mappings is introduced in the setting of 119887-metric spaces Existence and uniquenessof coincidence and common fixed points for such kind of mappings are investigated Results are supported with relevant examplesAt the end results are applied to find the solution of an integral equation

1 Introduction and Prelims

Fixed point theorems inmetric spaces and generalizedmetricspaces provide a tool to solve many problems and haveapplications in nonlinear analysis and in many other fieldsMost of the problems of applied mathematics reduce to solvea given equality which in turn may be reduced to find thefixed points of a certain mapping or the common fixedpoints of pairs of mappings In order to solve particularproblems researchers tried to generalize various contractionconditions auxiliary mappings and metric spaces Here inthis paper we will present common fixed point theoremsfor 120572-120573-contractive mappings in the framework of b-metricspace Therefore to have a clear understanding of the paperwe will discuss the b-metric space Geraghty type mappingsand 120572-admissible mappings step by step

The b-metric space or metric type space was introducedby Czerwik [1] in 1993 In this interesting paper Czerwik [1]generalized the Banach contraction principle in the contextof complete b-metric spaces After that many researchersreported the existence and uniqueness of fixed points ofvarious operators in the setting of b-metric spaces (see eg[2ndash13] and some references therein)

Let us have a look on definitions examples and proper-ties of b-metric space

Definition 1 (see [1 4]) Let X be a nonempty space and let119904 ge 1 be a given real number A functional119889 119883times119883 rarr [0infin)is said to be b-metric if the following conditions hold good

(1) 119889(119909 119910) = 0 if and only if 119909 = 119910(2) 119889(119909 119910) = 119889(119910 119909)(3) 119889(119909 119911) le 119904[119889(119909 119910) + 119889(119910 119911)] for all 119909 119910 and 119911 isin 119883If d satisfies all the above b-metric axioms then the pair(119883 119889) is called a b-metric or metric type spaceThe class of b-metric spaces is larger than that of metric

spaces since a b-metric is a metric when 119904 = 1 All the metricspaces are b-metric spaces but not vice versaWe can illustratethis with the help of the following example

Example 2 Let 119883 = 1199091 1199092 1199093 1199094 and 119889(1199091 1199092) = 119896 ge 2119889(1199091 1199093) = 119889(1199091 1199094) = 119889(1199092 1199093) = 119889(1199092 1199094) = 119889(1199093 1199094) = 1119889(119909119894 119909119895) = 119889(119909119895 119909119894) for 119894 119895 = 1 2 3 4 and 119889(119909119894 119909119894) = 0 119894 =1 2 3 4Then

119889 (119909119894 119909119895) le 1198962 [119889 (119909119894 119909119899) + 119889 (119909119899 119909119895)]for 119899 119894 119895 = 1 2 3 4

(1)

and if 119896 gt 2 the ordinary triangle inequality does not holdDefinition 3 (see [1]) Let (119883 119889) be a b-metric space

(a) A sequence 119909119899 in 119883 is called b-convergent if andonly if there exists 119909 isin 119883 such that 119889(119909119899 119909) rarr 0 as119899 rarr infin

(b) The sequence 119909119899 in 119883 is said to be b-Cauchy if andonly if 119889(119909119899 119909119898) rarr 0 as n119898 rarrinfin

HindawiAbstract and Applied AnalysisVolume 2017 Article ID 9718535 10 pageshttpsdoiorg10115520179718535

2 Abstract and Applied Analysis

The b-metric space (119883 119889) is called b-complete if every b-Cauchy sequence in119883 is b-convergent

The following lemmas are useful to prove our results

Lemma 4 (see [9]) Let (119883 119889) be a b-metric space and let 119909119899be a sequence in 119883 such that

lim119899rarrinfin

119889 (119909119899 119909119899+1) = 0 (2)

If 119909119899 is not a b-Cauchy sequence then there exist 120576 gt 0and two sequences 119898(119896) and 119899(119896) of positive integers suchthat the four sequences

119889 (119909119898(119896) 119909119899(119896)) 119889 (119909119898(119896) 119909119899(119896)+1) 119889 (119909119898(119896)+1 119909119899(119896)) 119889 (119909119898(119896)+1 119909119899(119896)+1)

(3)

exist and the following hold

120576 le lim119896rarrinfin

inf 119889 (119909119898(119896) 119909119899(119896))le lim119896rarrinfin

sup 119889 (119909119898(119896) 119909119899(119896)) le 120576119904 (4)

120576119904 le lim119896rarrinfin

inf 119889 (119909119898(119896) 119909119899(119896)+1)le lim119896rarrinfin

sup 119889 (119909119898(119896) 119909119899(119896)+1) le 1205761199042(5)

120576119904 le lim119896rarrinfin

inf 119889 (119909119898(119896)+1 119909119899(119896))le lim119896rarrinfin

sup 119889 (119909119898(119896)+1 119909119899(119896)) le 1205761199042(6)

1205761199042 le lim119896rarrinfin

inf 119889 (119909119898(119896)+1 119909119899(119896)+1)le lim119896rarrinfin

sup 119889 (119909119898(119896)+1 119909119899(119896)+1) le 1205761199043(7)

Lemma 5 (see [9]) Let (119883 119889) be a b-metric space and let 119909119899and 119910119899 be b-convergent to 119909 and 119910 respectivelyThen one has

11199042 119889 (119909 119910) le lim119899rarrinfin

inf 119889 (119909119899 119910119899) le lim119899rarrinfin

sup 119889 (119909119899 119910119899)le 1199042119889 (119909 119910)

(8)

In particular if 119909 = 119910 then one has lim119899rarrinfin119889(119909119899 119910119899) = 0Moreover for each 119911 isin 119883 one has

1119904 119889 (119909 119911) le lim119899rarrinfin

inf 119889 (119909119899 119911) le lim119899rarrinfin

sup 119889 (119909119899 119911)le 119904119889 (119909 119911)

(9)

Next we address briefly the concept of Geraghty typemappings

In 1973 Geraghty [14] generalized the Banach contrac-tion principle in the setting of complete metric spaces by

considering an auxiliary function This function is known asGeraghty type function or mapping Later on many authors[11 15ndash17] characterized the result of Geraghty in the contextof various metric spaces and proved many interesting resultsThe definition of this new class of mapping is as follows

Definition 6 Let 119861 denote the class of real functions 120573 [0 +infin) rarr [0 1) satisfying the condition120573 (119905119899) 997888rarr 1 implies 119905119899 997888rarr 0 (10)

For instance consider the function given by 120573(119905) = 119890minus2119905for 119905 gt 0 and 120573(0) isin [0 1) here 120573 isin 119861

In 1973 Geraghty generalized the Banach contractionprinciple in the following form

Theorem 7 (see [14]) Let (119883 119889) be a complete metric spaceand let 119865 119883 rarr 119883 be a self-map

Suppose that there exists 120573 isin 119861 such that

119889 (119865119909 119865119910) le 120573 (119889 (119909 119910)) 119889 (119909 119910) (11)

holds for all 119909 119910 isin 119883 Then 119865 has a unique fixed point 119911 isin 119883and for each 119909 isin 119883 the Picard sequence 119865119899119909 converges to 119911when 119899 rarr infin

In 2011 Dukic et al [16] introduced the Geraghty type 120573functions in b-metric space as follows

Definition 8 (see [16]) Let (119883 119889) be a b-metric space withgiven 119904 gt 1 Consider the class 119861119904 of real functions 120573 [0 +infin) rarr [0 1119904) satisfying the property

120573 (119905119899) 997888rarr 1119904 implies 119905119899 997888rarr 0 (12)

An example of a function in 119861119904 is given by 120573(119905) = 1119904119890minus119905for 119905 gt 0 and 120573(0) isin [0 1119904)

Lastly we discuss the 120572-admissible mappings examplesand alpha-admissibility for a pair of mappings

The concept of 120572-admissible mappings was introducedby Samet et al [18] in 2012 They established some fixedpoint theorems for such mappings in complete metric spacesand showed some examples and applications to ordinarydifferential equations Since then many researchers extendedthe idea and generalized fixed point results for single-valuedand multivalued 120572-admissible mappings in various abstractspaces (see eg [2 3 11 19 20] and more references in theliterature) The following definitions and examples reveal thebasics of 120572-admissible mappings

Definition 9 (see [18]) A self-mapping 119865 119883 rarr 119883 definedon a nonempty set 119883 is 120572-admissible if for all 119909 119910 isin 119883 onehas

120572 (119909 119910) ge 1 997904rArr 120572 (119865119909 119865119910) ge 1 (13)

where 120572 119883 times 119883 rarr [0infin) is a given function underconsideration

Abstract and Applied Analysis 3

Example 10 Let119883 = [0infin) and assume that 119865 119883 rarr 119883 and120572 119883 times 119883 rarr [0infin) by 119865119909 = radic119909 for all 119909 isin 119883 and120572 (119909 119910) =

119890119909minus119910 119909 ge 1199100 119909 lt 119910 (14)

Then F is 120572-admissible map

In 2014 a new notion of h-120572-admissible mapping wasintroduced by Rosa and Vetro [20] The definition of thisnotion is as follows

Definition 11 (see [20]) Let 119865 ℎ 119883 rarr 119883 be two self-mappings defined on a nonempty set 119883 And consider themap 120572 119883 times 119883 rarr [0infin) Then mapping F is called h-120572-admissible if for all 119909 119910 isin 119883

120572 (ℎ119909 ℎ119910) ge 1 implies 120572 (119865119909 119865119910) ge 1 (15)

Example 12 Let 119883 = [0infin) We will define the mapping 120572 119883 times 119883 rarr [0infin) by120572 (119909 119910) =

1 119909 ge 1199100 119909 lt 119910 (16)

and consider the mappings 119865 ℎ 119883 rarr 119883 by 119865(119909) = 119890119909and ℎ(119909) = 1199092 for all 119909 isin 119883 Then the mapping F is h-120572-admissible mapping

Allahyari et al [2] defined the 120572-regular space withrespect to some self-mapping defined on space as given below

Definition 13 (see [2]) Let (119883 119889) be a b-metric space Sup-pose that ℎ 119883 rarr 119883 and 120572 119883 times 119883 rarr [0infin) aretwo operators 119883 is 120572-regular with respect to ℎ if for everysequence 119909119899 isin 119883 120572(ℎ119909119899 ℎ119909119899+1) ge 1 for all 119899 isin N andℎ119909119899 rarr ℎ119909 isin ℎ(119883) as 119899 rarr infin then there exists a subsequenceℎ119909119899(119896) of ℎ119909119899 such that for all 119896 isin N 120572(ℎ119909119899(119896) ℎ119909) ge 1Definition 14 (see [11]) Let X be a nonempty set Then themap 120572 119883 times 119883 rarr [0infin) is called transitive if for 119906 V 119908 isin 119883one has

120572 (119906 V) ge 1120572 (V 119908) ge 1

dArr120572 (119906 119908) ge 1

(17)

In 2014 Sintunavarat [11] proved the fixed point theoremfor generalized 120572-120573-contractionmapping inmetric space andestablishedUlam-Hyers stability andwell-posedness via fixedpoint results The purpose of this paper is to define 120572-120573-contraction mapping for a pair of mappings and to provecoincidence point and common fixed point theorems in thecontext of b-metric space We will provide suitable examplesto support our results At the end we will discuss someapplications in the context of integral equations

2 Coincidence and CommonFixed Point Theorems

First of all we will present some definitions and results whichwill make our results easy to understand

Definition 15 Let 119865 and 119892 be two self-mappings defined ona nonempty set 119883 Then a point 119909 isin 119883 is called coincidencepoint of 119865 and 119892 if 119865119909 = 119892119909 Moreover if 119909 = 119865119909 = 119892119909 then119909 is called common fixed point of F and 119892Definition 16 Two self-mappings 119865 and 119892 defined on anonempty set119883 are weakly compatible if the maps commuteat each coincidence point that is if 119865119909 = 119892119909 for some 119909 isin 119883then 119865119892119909 = 119892119865119909

In 2014 Sintunavarat [11] defined the generalized 120572-120573-contraction mapping in metric spaces as follows

Definition 17 (see [11]) Let F be a self-mapping on anonempty set X and there exist two functions 120572 119883 times 119883 rarr[0infin) and 120573 isin 119861 We say that F is 120572-120573-contraction mappingif the following condition holds

[120572 (119909 119910) minus 1 + 120575lowast]119889(119865119909119865119910) le 120575120573(119889(119909119910))119889(119909119910) (18)

for all 119909 119910 isin 119883 where 1 lt 120575 le 120575lowastNow we will introduce our notions in the setting of b-

metric space

Definition 18 Let F and 119892 be two self-mappings defined onb-metric space (119883 119889) with given 119904 gt 1 and there exist twofunctions 120572 119883times119883 rarr [0infin) and 120573 isin 119861119904 We say that F is 120572-120573 (b)-contraction with respect to 119892 if the following conditionholds

[120572 (119892119909 119892119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120573(119889(119892119909119892119910))119889(119892119909119892119910) (19)

for all 119909 119910 isin 119883 where 1 lt 120575Next we present the coincidence and common fixed

point theorems for a new class of contraction in b-metricspace

Theorem 19 Let (119883 119889) be a complete b-metric space and let119865 119892 119883 rarr 119883 be two self-mappings such that 119865(119909) sube 119892(119909)and one of these two subsets of X is complete Suppose that Fis 120572-120573 (b)-contraction with respect to 119892mapping satisfying thefollowing conditions

(i) F is 119892-120572-admissible and 120572 is transitive mapping(ii) There exists 1199090 isin 119883 such that 120572(1198921199090 1198651199090) ge 1(iii) X is 120572-regular with respect to 119892Then F and 119892 have a coincidence point

Moreover if F and 119892 are weakly compatible and hypoth-esis has one more additional assumption(A1) either 120572(119906 V) ge 1 or 120572(V 119906) ge 1 whenever 119865119906 = 119892119906

and 119865V = 119892Vthen F and 119892 have a unique common fixed point

4 Abstract and Applied Analysis

Proof Let1199090 isin 119883 such that120572(1198921199090 1198651199090) ge 1 In order to provethat F and 119892 have a point of coincidence and using 119865(119909) sube119892(119909) define two sequences 119909119899 and 119910119899 in X such that

119910119899 = 119865119909119899 = 119892119909119899+1 forall119899 isin N (20)

Now if119910119899 = 119910119899+1 for any 119899 isin N then119892119909119899+1 = 119910119899 = 119910119899+1 =119865119909119899+1 and F and 119892 have a point of coincidenceWithout loss of generality one can suppose that119910119899 = 119910119899+1

for each 119899 isin NHence 119865 is 119892-120572-admissible and 120572(1198921199090 1198921199091) =120572(1198921199090 1198651199090) ge 1 Similarly 120572(1198921199091 1198921199092) = 120572(1198651199090 1198651199091) ge 1

By induction we can easily deduce

120572 (119892119909119899minus1 119892119909119899) ge 1 forall119899 isin N (21)

Step 1 We shall show that lim119899rarrinfin119889(119910119899+1 119910119899) = 0 Now itfollows from (19) and (21) for each 119899 isin N that

120575119889(119910119899+1 119910119899) = 120575119889(119865119909119899+1 119865119909119899)le [120572 (119892119909119899+1 119892119909119899) minus 1 + 120575]119889(119865119909119899+1 119865119909119899)le 120575120573(119889(119892119909119899+1 119892119909119899))119889(119892119909119899+1 119892119909119899)

(22)

This implies that

119889 (119910119899+1 119910119899) le 120573 (119889 (119892119909119899+1 119892119909119899)) 119889 (119892119909119899+1 119892119909119899)le 1119904 119889 (119910119899 119910119899minus1) forall119899 isin N (23)

Further from (23) it follows that

lim119899rarrinfin

119889 (119910119899+1 119910119899) = 0 (24)

Step 2 Next we show that 119910119899 is a Cauchy sequence in b-metric space 119883 On the contrary assume that 119910119899 is not aCauchy sequence Then there exists 120576 gt 0 and subsequencesof integers 119899119901 and119898119901 with 119899119901 gt 119898119901 ge 0 such that

119889 (119910119898119901 119910119899119901) gt 120576119889 (119910119898119901 119910119899119901minus1) lt 120576

forall119901 isin N(25)

Since 119899119901 gt 119898119901 and120572 is transitivemapping we can deduceeasily by using triangle inequality

120572 (119892119909119898119901+1 119892119909119899119901) ge 1 forall119901 isin N (26)

Now using (19) and (26) we can consider

120575119889(119910119898119901+1 119910119899119901 ) = 120575119889(119865119909119898119901+1 119865119909119899119901 )le [120572 (119892119909119898119901+1 119892119909119899119901) minus 1 + 120575]119889(119865119909119898119901+1 119865119909119899119901 )le 120575120573(119889(119892119909119898119901+1 119892119909119899119901 ))119889(119892119909119898119901+1 119892119909119899119901 )

(27)

This gives

119889 (119910119898119901+1 119910119899119901)le 120573 (119889 (119892119909119898119901+1 119892119909119899119901)) 119889 (119892119909119898119901+1 119892119909119899119901)le 120573 (119889 (119910119898119901 119910119899119901minus1)) 119889 (119910119898119901 119910119899119901minus1)le 120573 (119889 (119910119898119901 119910119899119901minus1)) 120576

(28)

119889 (119910119898119901+1 119910119899119901)120576 le 120573 (119889 (119910119898119901 119910119899119901minus1)) lt 1119904 (29)

with 119899119901 gt 119898119901 gt 119901 for all 119901 isin NHence by Lemma 4 and (29) we have

1119904 = 1120576 120576119904 le lim119901rarrinfin

inf 1120576 119889 (119910119898119901+1 119910119899119901)le lim119901rarrinfin

sup 1120576 119889 (119910119898119901+1 119910119899119901)le lim119901rarrinfin

inf 120573 (119889 (119910119898119901 119910119899119901minus1))le lim119901rarrinfin

sup120573 (119889 (119910119898119901 119910119899119901minus1)) le 1119904

(30)

that is

lim119901rarrinfin

120573 (119889 (119910119898119901 119910119899119901minus1)) = 1119904 (31)

Also 120573 isin 119861119904 implies that lim119901rarrinfin119889(119910119898119901 119910119899119901minus1) = 0However this is not possible as using (4) of Lemma 4 weget that

120576119904 le 1119904 119889 (119910119898119901 119910119899119901) le 119889 (119910119898119901 119910119899119901minus1) + 119889 (119910119899119901minus1 119910119899119901)997888rarr 0

(32)

as 119901 rarr infin which leads to contradicting (25)Therefore 119910119899 is a b-Cauchy sequence and by hypothesis

we can suppose that 119892(119883) is to be complete subspace ofX (proof can be derived in a similar manner when 119865(119883)is supposed to be complete) Then b-completeness of 119892(119883)implies that 119910119899 = 119865119909119899 = 119892119909119899+1 b-converges to a point119906 isin 119892(119883) where 119906 = 119892119911 for some 119911 isin 119883 or in other words

lim119899rarrinfin

119865119909119899 = lim119899rarrinfin

119892119909119899 = 119892119911 for some 119911 isin 119883 (33)

Step 3 Next we will prove that 119865119911 = 119892119911 Since X is 120572-regularwith respect to 119892 and using (33) we have

120572 (119892119909119899(119901) 119892119911) ge 1 forall119901 isin N (34)

Abstract and Applied Analysis 5

To show 119865119911 = 119892119911 we consider120575119889(119865119911119892119911) le 120575119904119889(119865119911119865119909119899(119901))+119904119889(119892119909119899(119901)+1 119892119911)

le 120575119904119889(119865119911119865119909119899(119901)) times 120575119904119889(119892119909119899(119901)+1119892119911)le [120572 (119892119911 119892119909119899(119901)) minus 1 + 120575]119904119889(119865119911119865119909119899(119901))times 120575119904119889(119892119909119899(119901)+1119892119911)

le 120575119904120573(119889(119892119911119892119909119899(119901)))119889(119892119911119892119909119899(119901)) times 120575119904119889(119892119909119899(119901)+1 119892119911)

(35)

This in turns implies that

119889 (119865119911 119892119911) le 119904120573 (119889 (119892119911 119892119909119899(119901))) 119889 (119892119911 119892119909119899(119901))+ 119904119889 (119892119909119899(119901)+1 119892119911)

1119904 119889 (119865119911 119892119911) le 120573 (119889 (119892119911 119892119909119899(119901))) 119889 (119892119911 119892119909119899(119901))+ 119889 (119892119909119899(119901)+1 119892119911)

lt 1119904 119889 (119892119911 119892119909119899(119901)) + 119889 (119892119909119899(119901)+1 119892119911)997888rarr 0 as 119899 997888rarr infin

(36)

Thus 119865119911 = 119892119911 = 119906 is a point of coincidence of F and 119892Step 4 Next we prove that F and 119892 have a common fixedpoint Firstly we claim that if 119865119906 = 119892119906 and 119865V = 119892V then119892119906 = 119892V By hypotheses 120572(119906 V) ge 1 or 120572(V 119906) ge 1 Supposethat 120572(119906 V) ge 1 and consider that

120575119889(119892119906119892V) = 120575119889(119865119906119865V) le [120572 (119892119906 119892V) minus 1 + 120575]119889(119865119906119865V)le 120575120573(119889(119892119906119892V))119889(119892119906119892V) (37)

Then we get that

119889 (119892119906 119892V) le 120573 (119889 (119892119906 119892V)) 119889 (119892119906 119892V)119889 (119892119906 119892V) le 119889 (119892119906 119892V) (38)

which is a contradictionTherefore we conclude that 119892119906 = 119892V If we take 120572(V 119906) ge1 then we also get the same resultSecondly If F and 119892 are weakly compatible then for any119906 isin 119883 if 119906 = 119865V = 119892V (because F and 119892 have a point

of coincidence proven earlier in Step 3) then using weakcompatibility of F and 119892 we get

119865119906 = 119865 (119892V) = 119892 (119865V) = 119892119906 (39)

Thus 119906 is a coincidence point of F and 119892 then using theresult in first part 119892119906 = 119892V which leads to 119865119906 = 119892119906 = 119892V =119865V = 119906

Therefore 119906 is a common fixed point of F and 119892We can prove the uniqueness of the common fixed point

of F and 119892 by making use of condition (19) and assumption(A1) of hypothesis The proof is very simple therefore we donot go through details

If we consider 119892 as identity mapping in the abovetheorem we deduce the following corollary

Corollary 20 Let (119883 119889) be a complete b-metric space and let119865 119883 rarr 119883 be continuous 120572-admissible mapping satisfying thefollowing condition

[120572 (119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120573(119889(119909119910))119889(119909119910) (40)

for all 119909 119910 isin 119883 and 120573 isin 119861119904 where 1 lt 120575If 120572 is transitive mapping and there exists 1199090 isin 119883 such that120572(1199090 1198651199090) ge 1 then F has a fixed point

Taking 120572(119909 119910) = 1 and 120575 = 1 in Corollary 20 we get thefollowing variant of Geraghty theorem

Corollary 21 Let (119883 119889) be a complete b-metric space and let119865 119883 rarr 119883 be self-mapping satisfying the following condition

119889 (119865119909 119865119910) le 120573 (119889 (119909 119910)) 119889 (119909 119910) (41)

for all 119909 119910 isin 119883 and 120573 isin 119861119904 Then F has a unique fixed point119911 isin 119883 and for each119909 isin 119883 the Picard sequence 119865119899119909 convergesto 119911 when 119899 rarr infin

The following lemma derived from [7] is very useful toprove our next theorem

Lemma 22 (see [7]) Let 119910119899 be a sequence in a metric typespace or b-metric space (X d) such that

119889 (119910119899+1 119910119899) le 120582119889 (119910119899 119910119899minus1) (42)

for some 120582 0 lt 120582 lt 1119904 and each 119899 = 1 2 Then 119910119899 is aCauchy sequence in X

Theorem 23 Let (119883 119889) be a complete b-metric space and be120572-regular with respect to 119892 And let 119865 119892 119883 rarr 119883 be two self-mappings where F is 119892-120572-admissible and 120572 is transitive andfor 1199090 isin 119883 one has 120572(1198921199090 1198651199090) ge 1 If 119865(119909) sube 119892(119909) and oneof these two subsets of 119883 is complete suppose that there exists120582 isin [0 1119904) such that

[120572 (119892119909 119892119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119865119892119909119910) (43)

for all 119909 119910 isin 119883 and 1 lt 120575Consider

119872(119865 119892 119909 119910) = max119889 (119892119909 119892119910) 119889 (119892119909 119865119909) 119889 (119892119910 119865119910) 119889 (119892119909 119865119910) + 119889 (119892119910 119865119909)2119904

(44)

Then F and 119892 have a coincidence point

Moreover if F and 119892 are weakly compatible and hypoth-esis has one more additional assumption

(A1) either 120572(119906 V) ge 1 or 120572(V 119906) ge 1 whenever 119865119906 = 119892119906and 119865V = 119892V

then F and 119892 have a unique common fixed point

6 Abstract and Applied Analysis

Proof Let 1199090 isin 119883 be arbitrary and using condition119865(119883) sube 119892(119883) we can easily construct a Jungck sequence 119910119899satisfying

119910119899 = 119865119909119899 = 119892119909119899+1 forall119899 isin N (45)

Now if119910119899 = 119910119899+1 for any 119899 isin N then119892119909119899+1 = 119910119899 = 119910119899+1 =119865119909119899+1 and F and 119892 have a point of coincidenceWithout loss of generality we assume that 119910119899 = 119910119899+1 for

each 119899 isin NHence F is 119892-120572-admissible and 120572(1198921199090 1198921199091) = 120572(11989211990901198651199090) ge 1 Similarly 120572(1198921199091 1198921199092) = 120572(1198651199090 1198651199091) ge 1 By

induction we can easily deduce

120572 (119892119909119899minus1 119892119909119899) ge 1 forall119899 isin N (46)

Step 1 We shall show that 119910119899 is a b-Cauchy sequence Nowtaking (43) and (46) we obtain for each 119899 isin N

120575119889(119910119899+1 119910119899) = 120575119889(119865119909119899+1 119865119909119899)le [120572 (119892119909119899+1 119892119909119899) minus 1 + 120575]119889(119865119909119899+1 119865119909119899)le 120575120582119872(119865119892119909119899+1119909119899)

(47)

This implies that

119889 (119910119899+1 119910119899) le 120582119872(119865 119892 119909119899+1 119909119899) (48)

or

119889 (119910119899+1 119910119899) le 120582max119889 (119892119909119899+1 119892119909119899) 119889 (119892119909119899+1 119865119909119899+1) 119889 (119892119909119899 119865119909119899) 119889 (119892119909119899+1 119865119909119899) + 119889 (119892119909119899 119865119909119899+1)2119904

le 120582max119889 (119910119899 119910119899minus1) 119889 (119910119899 119910119899+1) 119889 (119910119899minus1 119910119899) 119889 (119910119899 119910119899) + 119889 (119910119899minus1 119910119899+1)2119904

le 120582max119889 (119910119899 119910119899minus1) 119889 (119910119899minus1 119910119899) + 119889 (119910119899 119910119899+1)2

(49)

Now if max 119889(119910119899 119910119899minus1) (119889(119910119899minus1 119910119899) + 119889(119910119899 119910119899+1))2 =(119889(119910119899minus1 119910119899)+119889(119910119899 119910119899+1))2 then 119889(119910119899 119910119899minus1) lt (119889(119910119899minus1 119910119899)+119889(119910119899 119910119899+1))2 lt 119889(119910119899+1 119910119899)Then (49) implies that

119889 (119910119899+1 119910119899) le 120582119889 (119910119899+1 119910119899) (50)

which is impossible as 120582 lt 1Therefore we deduce that max 119889(119910119899 119910119899minus1) (119889(119910119899minus1 119910119899)+119889(119910119899 119910119899+1))2 = 119889(119910119899 119910119899minus1)This in turn implies that

119889 (119910119899+1 119910119899) le 120582119889 (119910119899 119910119899minus1) (51)

Using Lemma 22 we obtain that 119910119899 is a b-Cauchysequence and by hypothesis we can suppose that 119892(119883) is tobe complete subspace of 119883 (the proof when 119865(119883) is similar)Then b-completeness of 119892(119883) implies that 119910119899 = 119865119909119899 =119892119909119899+1 b-converges to a point 119906 isin 119892(119883) where 119906 = 119892119911 forsome 119911 isin 119883 or in other words

lim119899rarrinfin

119865119909119899 = lim119899rarrinfin

119892119909119899 = 119892119911 for some 119911 isin 119883 (52)

Step 2 Next we will prove that 119865119911 = 119892119911 Since119883 is 120572-regularwith respect to 119892 and using (52) we have

120572 (119892119909119899(119901) 119892119911) ge 1 forall119901 isin N (53)

To show 119865119911 = 119892119911 we consider120575119889(119865119909119899(119901) 119865119911)le [120572 (119892119909119899(119901) 119892119911) minus 1 + 120575]119889(119865119909119899(119901) 119865119911) le 120575120582119872(119865119892119909119899(119901) 119911)

(54)

This implies that

119889 (119865119909119899(119901) 119865119911) le 120582119872(119865 119892 119909119899(119901) 119911) (55)

or

119889 (119865119909119899(119901) 119865119911) le 120582max119889 (119892119909119899(119901) 119892119911) 119889 (119892119909119899(119901) 119865119909119899(119901)) 119889 (119892119911 119865119911) 119889 (119892119909119899(119901) 119865119911) + 119889 (119892119911 119865119909119899(119901))2119904

(56)

Also119865119909119899(119901) rarr 119892119911 and 119892119909119899(119901) rarr 119892119911 as 119899 rarr infin implyingthat

119889 (119892119909119899(119901) 119865119909119899(119901)) = 119904119889 (119892119909119899(119901) 119892119911) + 119904119889 (119892119911 119865119909119899(119901))997888rarr 0 as 119899 997888rarr infin (57)

Then we have only two cases

Case 1

119889 (119865119909119899(119901) 119865119911) le 120582119889 (119892119911 119865119911)le 120582119904 (119889 (119892119911 119865119909119899(119901)) + 119889 (119865119909119899(119901) 119865119911))

(1 minus 120582119904) 119889 (119865119909119899(119901) 119865119911) le 120582119904119889 (119892119911 119865119909119899(119901)) 997888rarr 0as 119899 997888rarr infin

(58)

Abstract and Applied Analysis 7

Since 1 minus 120582119904 gt 0 it follows that 119865119909119899(119901) rarr 119865119911Case 2

119889 (119865119909119899(119901) 119865119911) le 120582119889 (119892119909119899(119901) 119865119911)2119904le 120582119904(119889 (119892119909119899(119901) 119865119909119899(119901)) + 119889 (119865119909119899(119901) 119865119911)2119904 )

(1 minus 1205822)119889 (119865119909119899(119901) 119865119911) le 1205822119889 (119892119909119899(119901) 119865119909119899(119901)) 997888rarr 0as 119899 997888rarr infin

(59)

Since 1 minus 1205822 gt 0 it follows that 119865119909119899(119901) rarr 119865119911Uniqueness of the limit of a sequence implies that 119865119911 =119892119911Using conditions (A1) and weak compatibility of F and 119892

of hypothesis we can easily prove that F and 119892 have a uniquecommon fixed point

From the above theorem one can deduce the followingcorollary easily

Corollary 24 Let (119883 119889) be a complete b-metric space and 119865 119883 rarr 119883 is continuous and 120572-admissible Suppose that thereexists 120582 isin [0 1119904) such that

[120572 (119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119909119910) (60)

for all 119909 119910 isin 119883 and 1 lt 120575Consider

119872(119909 119910) = max119889 (119909 119910) 119889 (119909 119865119909) 119889 (119910 119865119910) 119889 (119909 119865119910) + 119889 (119910 119865119909)

2119904 (61)

If 120572 is a transitive mapping and there exists 1199090 isin 119883 suchthat 120572(1199090 1198651199090) ge 1 then 119865 has a fixed point

Taking 120572(119909 119910) = 1 and 120575 = 1 we get the following variantof Corollary 312 of [7]

Corollary 25 Let (119883 119889) be a complete b-metric space and let119865 119883 rarr 119883 be self-mapping Suppose that there exists 120582 isin[0 1119904) such that for all 119909 119910 isin 119883119889 (119865119909 119865119910) le 120582119872(119909 119910) (62)

where

119872(119909 119910) = max119889 (119909 119910) 119889 (119909 119865119909) 119889 (119910 119865119910) 119889 (119909 119865119910) + 119889 (119910 119865119909)

2119904 (63)

Then F has a unique fixed point 119911 isin 119883

In what follows we furnish illustrative examples whereinone demonstrates Theorems 19 and 23 on the existence anduniqueness of a common fixed point

Example 26 Let119883 = 0 1 3 be a b-metric space withmetricd given by 119889(119909 119910) = (119909 minus 119910)2 with 119904 = 2 Consider themappings 119865 119892 119883 rarr 119883 defined by 119865(0) = 1 119865(1) = 1and 119865(3) = 1 and 119892 by 119892(0) = 1 119892(1) = 1 and 119892(3) = 3 Letus take 120573 isin 119861119904 as 120573(119905) = 12 for 119905 gt 0 and 120573(0) isin [0 12)

And 120572 119883 times 119883 rarr [0infin) by

120572 (119909 119910) = 1 when 119909 119910 ge 00 otherwise

(64)

then one can examine easily that

119889 (1198650 1198651) = 119889 (1 1) = 0120573 (119889 (1198920 1198921)) 119889 (1198920 1198921) = 120573 (119889 (1 1)) 119889 (1 1) = 0

119889 (1198650 1198653) = 119889 (1 1) = 0120573 (119889 (1198920 1198923)) 119889 (1198920 1198923) = 120573 (119889 (1 3)) 119889 (1 3) = 2

119889 (1198651 1198653) = 119889 (1 1) = 0120573 (119889 (1198921 1198923)) 119889 (1198921 1198923) = 120573 (119889 (1 3)) 119889 (1 3) = 2

(65)

Thus in all cases we get

[120572 (119892119909 119892119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120573(119889(119892119909119892119910))119889(119892119909119892119910) (66)

for all 119909 119910 isin 119883 where 1 lt 120575Also we can check that F and 119892 satisfy all the other

assumptions of Theorem 19 and thus the pair F and 119892 has aunique common fixed point 119909 = 1Example 27 Let119883 = 0 1 3 be a b-metric space with metricd given by 119889(119909 119910) = (119909 minus 119910)2 with 119904 = 2 Consider themappings 119865 119892 119883 rarr 119883 defined by 119865(0) = 0 119865(1) = 1and 119865(3) = 0 and 119892 by 119892(0) = 3 119892(1) = 1 and 119892(3) = 0 Letus take 120582 isin [0 12)

And 120572 119883 times 119883 rarr [0infin) by

120572 (119909 119910) = 1 when 119909 119910 ge 00 otherwise

(67)

Now we verify Theorem 23 by considering the followingcases

8 Abstract and Applied Analysis

Case 1 (take points 0 and 1)

119889 (1198650 1198651) = 119889 (0 1) = 1119889 (1198920 1198921) = 119889 (3 1) = 4119889 (1198920 1198650) = 119889 (3 0) = 9119889 (1198921 1198651) = 119889 (1 1) = 0119889 (1198920 1198651) + 119889 (1198921 1198650)

2 sdot 2 = 119889 (3 1) + 119889 (1 0)4 = 4 + 14= 54

119872 (119865 119892 0 1) = max119889 (1198920 1198921) 119889 (1198920 1198650) 119889 (1198921 1198651) 119889 (1198920 1198651) + 119889 (1198921 1198650)4 = 4 9 0 54= 9

(68)

Case 2 (take points 1 and 3)

119889 (1198651 1198653) = 119889 (1 1) = 0119872 (119865 119892 1 3) = max119889 (1198921 1198923) 119889 (1198921 1198651) 119889 (1198923 1198653) 119889 (1198921 1198653) + 119889 (1198923 1198651)4 = 1 0 0 24= 1

(69)

Case 3 (take points 0 and 3)

119889 (1198650 1198653) = 119889 (0 0) = 0119872 (119865 119892 0 3) = max119889 (1198920 1198923) 119889 (1198920 1198650) 119889 (1198923 1198653) 119889 (1198920 1198653) + 119889 (1198923 1198650)4 = 9 9 0 94= 9

(70)

Thus in all cases F and 119892 satisfy all the assumptions ofTheorem 23 and thus the pair F and 119892 has a unique commonfixed point 119909 = 13 An Application to an Integral Equation

This section deals with the applications of results proven inthe previous section Here we will investigate the solution ofintegral equation through our results

Consider the following integral equation

119909 (119905) = int10119870 (119905 119903 119909 (119903)) 119889119903 (71)

where119870 [0 1] times [0 1] timesR rarr R

Let 119883 = 119862[0 1] be the set of continuous real functionsdefined on [0 1] Define the b-metric by 119889(119909(119905) 119910(119905)) =max119905isin[01](|119909(119905)| + |119910(119905)|)119901 for all 119909 119910 isin 119883

Consider 119901 gt 1 Then (119883 119889) is a complete b-metric spacewith the constant 119904 = 2119901minus1119865(119909(119905)) = int1

0119870(119905 119903 119909(119903))119889119903 for all 119909 isin 119883 and for all 119905 isin[0 1] Then the existence of a solution to (71) is equivalent to

the existence of a fixed point of F Nowwe prove the followingresult

Theorem 28 Let us suppose that the following hypotheseshold

(i) 119870 [0 1] times [0 1] timesR rarr R is continuous

(ii) For all 119905 119903 isin [0 1] there exists a continuous operator120585 [0 1] times [0 1] rarr R such that

|119870 (119905 119903 119909 (119903))| + 1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816lt 1205821119901120585 (119905 119903) (|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)

sup119905isin[01]

int10120585 (119905 119903) 119889119903 le 1

(72)

where 0 lt 120582 lt 1119904(iii) There exists a function 120583 R2 rarr R such that for all119905 isin 119868 and for all 119886 119887 isin R with 120583(119886 119887) ge 0 we have

120583(1199091 (119905) int10119870 (119905 119903 119909 (119903)) 119889119903) ge 0

for 119909 isin 119862 (119868) forall119905 isin 119868120583 (119909 (119905) 119910 (119905)) ge 0

120583 (int10119870 (119905 119903 119909 (119903)) int1

0119870(119905 119903 119910 (119903)) 119889119903) ge 0

forall119905 isin 119868 forall119909 119910 isin 119862 (119868)

(73)

Then the integral equation (71) has a unique solution 119909 isin119883Proof We define 120572 119862(119868) times 119862(119868) rarr [0infin) by

120572 (119909 119910) = 1 when 120583 (119909 (119905) 119910 (119905)) ge 00 otherwise

(74)

Then for all 119909 119910 isin 119862(119868) we have 120572(119909 119910) = 1 and120572(119910 119911) = 1 implying that 120572(119909 119911) = 1 for all 119909 119910 119911 isin 119862(119868)which proves that 120572 is a transitive mapping

If 120572(119909 119910) = 1 for all 119909 119910 isin 119862(119868) then 120583(119909(119905) 119910(119905)) ge 0From (iii) we have 120583(119865119909(119905) 119865119910(119905)) ge 0 and so 120572(119865119909 119865119910) = 1Thus F is 120572-admissible

From (iii) there exists 1199091 isin 119862(119868) such that 120572(1199091 1198651199091) = 1

Abstract and Applied Analysis 9

Also we observe from (72) that

(|119865119909 (119905)| + 1003816100381610038161003816119865119910 (119905)1003816100381610038161003816)119901= (100381610038161003816100381610038161003816100381610038161003816int

1

0119870 (119905 119903 119909 (119903)) 119889119903100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816int1

0119870(119905 119903 119910 (119903)) 119889119903100381610038161003816100381610038161003816100381610038161003816)

119901

le (int10|119870 (119905 119903 119909 (119903))| 119889119903 + int1

0

1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816 119889119903)119901

le (int10(|119870 (119905 119903 119909 (119903))| + 1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816) 119889119903)

119901

le (int10(1205821119901120585 (119905 119903) (|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)) 119889119903)

119901

= (int10(1205821119901120585 (119905 119903) ((|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)119901)1119901)119889119903)

119901

le (int101205821119901120585 (119905 119903) 1198891119901 (119909 (119903) 119910 (119903)) 119889119903)119901

= 120582119889 (119909 (119905) 119910 (119905)) (int10120585 (119905 119903) 119889119903)119901

le 120582119889 (119909 (119905) 119910 (119905)) le 120582119872(119909 (119905) 119910 (119905))

(75)

which in turn implies that [120572(119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119909119910)Now all the conditions of Corollary 24 hold and F has a

unique fixed point 119909 isin 119883 which means that 119909 is the uniquesolution for the integral equation (71)

4 Conclusion

To conclude we can assert that our results are novelinteresting and generalized while considering the alpha-admissible Geraghty type mappings These results extendimprove unify and generalize many theorems based onalpha-admissible mappings in 119887-metric spaces Examples arepresented in a simplest form and illustrate the theoremsApplication to integral equation adds value to our research

Conflicts of Interest

All the authors of this article declare that they have noconflicts of interest regarding the publication of this article

Authorsrsquo Contributions

All authors have equal contribution in writing this paper Allauthors read and approved the final paper

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] R Allahyari R Arab and A S Haghighi ldquoFixed pointsof admissible almost contractive type mappings on b-metric

spaces with an application to quadratic integral equationsrdquoJournal of Inequalities and Applications vol 2015 no 32 18pages 2015

[3] M-F Bota E Karapinar andOMlesnite ldquoUlam-hyers stabilityresults for fixed point problems via 120572-120595-contractive mappingin (b)-metric spacerdquo Abstract and Applied Analysis vol 2013Article ID 825293 6 pages 2013

[4] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacerdquo Functional Analysis vol 30 pp 26ndash37 1989

[5] C Chen J Dong and C Zhu ldquoSome fixed point theorems inb-metric-like spacesrdquo Fixed Point Theory and Applications vol2015 no 122 10 pages 2015

[6] H-SDingM Imdad S Radenovic and J Vujakovic ldquoOn somefixed point results in b-metric rectangular and b-rectangularmetric spacesrdquo Arab Journal of Mathematical Sciences vol 22(2016) pp 151ndash164 2015

[7] S Radenovic M Jovanovic and Z Kadelburg ldquoCommon fixedpoint results in metric-type spacesrdquo Fixed Point Theory andApplications vol 2010 article 978121 15 pages 2010

[8] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 article 315398 2010

[9] J R Roshan V Parvaneh and Z Kadelburg ldquoCommon fixedpoint theorems for weakly isotone increasing mappings inordered b-metric spacesrdquo Journal of Nonlinear Science and itsApplications JNSA vol 7 no 4 pp 229ndash245 2014

[10] W Sintunavarat S Plubtieng and P Katchang ldquoFixed pointresult and applications on a b-metric space endowed with anarbitrary binary relationrdquo Fixed Point Theory and Applicationsvol 2013 article 296 2013

[11] W Sintunavarat ldquoGeneralized Ulam-Hyers stability well-posedness and limit shadowing of fixed point problems for 120572-120573-contraction mapping in metric spacesrdquo The Scientific WorldJournal vol 2014 Article ID 569174 7 pages 2014

[12] R J Shahkoohi and A Razani ldquoSome fixed point theoremsfor rational Geraghty contractivemappings in ordered b-metricspacesrdquo Journal of Inequalities and Applications vol 373 p 232014

[13] S L Singh and B Prasad ldquoSome coincidence theorems andstability of iterative proceduresrdquoComputersampMathematics withApplications vol 55 no 11 pp 2512ndash2520 2008

[14] M A Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[15] S Cho J Bae and E Karapınar ldquoFixed point theorems for 120572-Geraghty contraction type maps in metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 329 11 pages 2013

[16] D Dukic Z Kadelburg and S Radenovic ldquoFixed points ofGeraghty-type mappings in various generalized metric spacesrdquoAbstract and Applied Analysis vol 2011 Article ID 561245 13pages 2011

[17] M E Gordji M Ramezani Y J Cho and S Pirbavafa ldquoAgeneralization ofGeraghtyrsquos theorem in partially orderedmetricspaces and applications to ordinary differential equationsrdquoFixed Point Theory and Applications vol 2012 article 74 2012

[18] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

10 Abstract and Applied Analysis

[19] A Felhi S Sahmim and H Aydi ldquoUlam-hyers stability andwell-posedness of fixed point problems for 120572-120582-metric spacesrdquoFixed Point Theory and Applications vol 2016 no 1 20 pages2016

[20] V L Rosa and P Vetro ldquoCommon fixed points for 120572-120595-120601-contractions in generalized metric spacesrdquo Nonlinear AnalysisModelling and Control vol 19 no 1 pp 43ndash54 2014

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Abstract and Applied Analysis 3

Example 10 Let119883 = [0infin) and assume that 119865 119883 rarr 119883 and120572 119883 times 119883 rarr [0infin) by 119865119909 = radic119909 for all 119909 isin 119883 and120572 (119909 119910) =

119890119909minus119910 119909 ge 1199100 119909 lt 119910 (14)

Then F is 120572-admissible map

In 2014 a new notion of h-120572-admissible mapping wasintroduced by Rosa and Vetro [20] The definition of thisnotion is as follows

Definition 11 (see [20]) Let 119865 ℎ 119883 rarr 119883 be two self-mappings defined on a nonempty set 119883 And consider themap 120572 119883 times 119883 rarr [0infin) Then mapping F is called h-120572-admissible if for all 119909 119910 isin 119883

120572 (ℎ119909 ℎ119910) ge 1 implies 120572 (119865119909 119865119910) ge 1 (15)

Example 12 Let 119883 = [0infin) We will define the mapping 120572 119883 times 119883 rarr [0infin) by120572 (119909 119910) =

1 119909 ge 1199100 119909 lt 119910 (16)

and consider the mappings 119865 ℎ 119883 rarr 119883 by 119865(119909) = 119890119909and ℎ(119909) = 1199092 for all 119909 isin 119883 Then the mapping F is h-120572-admissible mapping

Allahyari et al [2] defined the 120572-regular space withrespect to some self-mapping defined on space as given below

Definition 13 (see [2]) Let (119883 119889) be a b-metric space Sup-pose that ℎ 119883 rarr 119883 and 120572 119883 times 119883 rarr [0infin) aretwo operators 119883 is 120572-regular with respect to ℎ if for everysequence 119909119899 isin 119883 120572(ℎ119909119899 ℎ119909119899+1) ge 1 for all 119899 isin N andℎ119909119899 rarr ℎ119909 isin ℎ(119883) as 119899 rarr infin then there exists a subsequenceℎ119909119899(119896) of ℎ119909119899 such that for all 119896 isin N 120572(ℎ119909119899(119896) ℎ119909) ge 1Definition 14 (see [11]) Let X be a nonempty set Then themap 120572 119883 times 119883 rarr [0infin) is called transitive if for 119906 V 119908 isin 119883one has

120572 (119906 V) ge 1120572 (V 119908) ge 1

dArr120572 (119906 119908) ge 1

(17)

In 2014 Sintunavarat [11] proved the fixed point theoremfor generalized 120572-120573-contractionmapping inmetric space andestablishedUlam-Hyers stability andwell-posedness via fixedpoint results The purpose of this paper is to define 120572-120573-contraction mapping for a pair of mappings and to provecoincidence point and common fixed point theorems in thecontext of b-metric space We will provide suitable examplesto support our results At the end we will discuss someapplications in the context of integral equations

2 Coincidence and CommonFixed Point Theorems

First of all we will present some definitions and results whichwill make our results easy to understand

Definition 15 Let 119865 and 119892 be two self-mappings defined ona nonempty set 119883 Then a point 119909 isin 119883 is called coincidencepoint of 119865 and 119892 if 119865119909 = 119892119909 Moreover if 119909 = 119865119909 = 119892119909 then119909 is called common fixed point of F and 119892Definition 16 Two self-mappings 119865 and 119892 defined on anonempty set119883 are weakly compatible if the maps commuteat each coincidence point that is if 119865119909 = 119892119909 for some 119909 isin 119883then 119865119892119909 = 119892119865119909

In 2014 Sintunavarat [11] defined the generalized 120572-120573-contraction mapping in metric spaces as follows

Definition 17 (see [11]) Let F be a self-mapping on anonempty set X and there exist two functions 120572 119883 times 119883 rarr[0infin) and 120573 isin 119861 We say that F is 120572-120573-contraction mappingif the following condition holds

[120572 (119909 119910) minus 1 + 120575lowast]119889(119865119909119865119910) le 120575120573(119889(119909119910))119889(119909119910) (18)

for all 119909 119910 isin 119883 where 1 lt 120575 le 120575lowastNow we will introduce our notions in the setting of b-

metric space

Definition 18 Let F and 119892 be two self-mappings defined onb-metric space (119883 119889) with given 119904 gt 1 and there exist twofunctions 120572 119883times119883 rarr [0infin) and 120573 isin 119861119904 We say that F is 120572-120573 (b)-contraction with respect to 119892 if the following conditionholds

[120572 (119892119909 119892119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120573(119889(119892119909119892119910))119889(119892119909119892119910) (19)

for all 119909 119910 isin 119883 where 1 lt 120575Next we present the coincidence and common fixed

point theorems for a new class of contraction in b-metricspace

Theorem 19 Let (119883 119889) be a complete b-metric space and let119865 119892 119883 rarr 119883 be two self-mappings such that 119865(119909) sube 119892(119909)and one of these two subsets of X is complete Suppose that Fis 120572-120573 (b)-contraction with respect to 119892mapping satisfying thefollowing conditions

(i) F is 119892-120572-admissible and 120572 is transitive mapping(ii) There exists 1199090 isin 119883 such that 120572(1198921199090 1198651199090) ge 1(iii) X is 120572-regular with respect to 119892Then F and 119892 have a coincidence point

Moreover if F and 119892 are weakly compatible and hypoth-esis has one more additional assumption(A1) either 120572(119906 V) ge 1 or 120572(V 119906) ge 1 whenever 119865119906 = 119892119906

and 119865V = 119892Vthen F and 119892 have a unique common fixed point

4 Abstract and Applied Analysis

Proof Let1199090 isin 119883 such that120572(1198921199090 1198651199090) ge 1 In order to provethat F and 119892 have a point of coincidence and using 119865(119909) sube119892(119909) define two sequences 119909119899 and 119910119899 in X such that

119910119899 = 119865119909119899 = 119892119909119899+1 forall119899 isin N (20)

Now if119910119899 = 119910119899+1 for any 119899 isin N then119892119909119899+1 = 119910119899 = 119910119899+1 =119865119909119899+1 and F and 119892 have a point of coincidenceWithout loss of generality one can suppose that119910119899 = 119910119899+1

for each 119899 isin NHence 119865 is 119892-120572-admissible and 120572(1198921199090 1198921199091) =120572(1198921199090 1198651199090) ge 1 Similarly 120572(1198921199091 1198921199092) = 120572(1198651199090 1198651199091) ge 1

By induction we can easily deduce

120572 (119892119909119899minus1 119892119909119899) ge 1 forall119899 isin N (21)

Step 1 We shall show that lim119899rarrinfin119889(119910119899+1 119910119899) = 0 Now itfollows from (19) and (21) for each 119899 isin N that

120575119889(119910119899+1 119910119899) = 120575119889(119865119909119899+1 119865119909119899)le [120572 (119892119909119899+1 119892119909119899) minus 1 + 120575]119889(119865119909119899+1 119865119909119899)le 120575120573(119889(119892119909119899+1 119892119909119899))119889(119892119909119899+1 119892119909119899)

(22)

This implies that

119889 (119910119899+1 119910119899) le 120573 (119889 (119892119909119899+1 119892119909119899)) 119889 (119892119909119899+1 119892119909119899)le 1119904 119889 (119910119899 119910119899minus1) forall119899 isin N (23)

Further from (23) it follows that

lim119899rarrinfin

119889 (119910119899+1 119910119899) = 0 (24)

Step 2 Next we show that 119910119899 is a Cauchy sequence in b-metric space 119883 On the contrary assume that 119910119899 is not aCauchy sequence Then there exists 120576 gt 0 and subsequencesof integers 119899119901 and119898119901 with 119899119901 gt 119898119901 ge 0 such that

119889 (119910119898119901 119910119899119901) gt 120576119889 (119910119898119901 119910119899119901minus1) lt 120576

forall119901 isin N(25)

Since 119899119901 gt 119898119901 and120572 is transitivemapping we can deduceeasily by using triangle inequality

120572 (119892119909119898119901+1 119892119909119899119901) ge 1 forall119901 isin N (26)

Now using (19) and (26) we can consider

120575119889(119910119898119901+1 119910119899119901 ) = 120575119889(119865119909119898119901+1 119865119909119899119901 )le [120572 (119892119909119898119901+1 119892119909119899119901) minus 1 + 120575]119889(119865119909119898119901+1 119865119909119899119901 )le 120575120573(119889(119892119909119898119901+1 119892119909119899119901 ))119889(119892119909119898119901+1 119892119909119899119901 )

(27)

This gives

119889 (119910119898119901+1 119910119899119901)le 120573 (119889 (119892119909119898119901+1 119892119909119899119901)) 119889 (119892119909119898119901+1 119892119909119899119901)le 120573 (119889 (119910119898119901 119910119899119901minus1)) 119889 (119910119898119901 119910119899119901minus1)le 120573 (119889 (119910119898119901 119910119899119901minus1)) 120576

(28)

119889 (119910119898119901+1 119910119899119901)120576 le 120573 (119889 (119910119898119901 119910119899119901minus1)) lt 1119904 (29)

with 119899119901 gt 119898119901 gt 119901 for all 119901 isin NHence by Lemma 4 and (29) we have

1119904 = 1120576 120576119904 le lim119901rarrinfin

inf 1120576 119889 (119910119898119901+1 119910119899119901)le lim119901rarrinfin

sup 1120576 119889 (119910119898119901+1 119910119899119901)le lim119901rarrinfin

inf 120573 (119889 (119910119898119901 119910119899119901minus1))le lim119901rarrinfin

sup120573 (119889 (119910119898119901 119910119899119901minus1)) le 1119904

(30)

that is

lim119901rarrinfin

120573 (119889 (119910119898119901 119910119899119901minus1)) = 1119904 (31)

Also 120573 isin 119861119904 implies that lim119901rarrinfin119889(119910119898119901 119910119899119901minus1) = 0However this is not possible as using (4) of Lemma 4 weget that

120576119904 le 1119904 119889 (119910119898119901 119910119899119901) le 119889 (119910119898119901 119910119899119901minus1) + 119889 (119910119899119901minus1 119910119899119901)997888rarr 0

(32)

as 119901 rarr infin which leads to contradicting (25)Therefore 119910119899 is a b-Cauchy sequence and by hypothesis

we can suppose that 119892(119883) is to be complete subspace ofX (proof can be derived in a similar manner when 119865(119883)is supposed to be complete) Then b-completeness of 119892(119883)implies that 119910119899 = 119865119909119899 = 119892119909119899+1 b-converges to a point119906 isin 119892(119883) where 119906 = 119892119911 for some 119911 isin 119883 or in other words

lim119899rarrinfin

119865119909119899 = lim119899rarrinfin

119892119909119899 = 119892119911 for some 119911 isin 119883 (33)

Step 3 Next we will prove that 119865119911 = 119892119911 Since X is 120572-regularwith respect to 119892 and using (33) we have

120572 (119892119909119899(119901) 119892119911) ge 1 forall119901 isin N (34)

Abstract and Applied Analysis 5

To show 119865119911 = 119892119911 we consider120575119889(119865119911119892119911) le 120575119904119889(119865119911119865119909119899(119901))+119904119889(119892119909119899(119901)+1 119892119911)

le 120575119904119889(119865119911119865119909119899(119901)) times 120575119904119889(119892119909119899(119901)+1119892119911)le [120572 (119892119911 119892119909119899(119901)) minus 1 + 120575]119904119889(119865119911119865119909119899(119901))times 120575119904119889(119892119909119899(119901)+1119892119911)

le 120575119904120573(119889(119892119911119892119909119899(119901)))119889(119892119911119892119909119899(119901)) times 120575119904119889(119892119909119899(119901)+1 119892119911)

(35)

This in turns implies that

119889 (119865119911 119892119911) le 119904120573 (119889 (119892119911 119892119909119899(119901))) 119889 (119892119911 119892119909119899(119901))+ 119904119889 (119892119909119899(119901)+1 119892119911)

1119904 119889 (119865119911 119892119911) le 120573 (119889 (119892119911 119892119909119899(119901))) 119889 (119892119911 119892119909119899(119901))+ 119889 (119892119909119899(119901)+1 119892119911)

lt 1119904 119889 (119892119911 119892119909119899(119901)) + 119889 (119892119909119899(119901)+1 119892119911)997888rarr 0 as 119899 997888rarr infin

(36)

Thus 119865119911 = 119892119911 = 119906 is a point of coincidence of F and 119892Step 4 Next we prove that F and 119892 have a common fixedpoint Firstly we claim that if 119865119906 = 119892119906 and 119865V = 119892V then119892119906 = 119892V By hypotheses 120572(119906 V) ge 1 or 120572(V 119906) ge 1 Supposethat 120572(119906 V) ge 1 and consider that

120575119889(119892119906119892V) = 120575119889(119865119906119865V) le [120572 (119892119906 119892V) minus 1 + 120575]119889(119865119906119865V)le 120575120573(119889(119892119906119892V))119889(119892119906119892V) (37)

Then we get that

119889 (119892119906 119892V) le 120573 (119889 (119892119906 119892V)) 119889 (119892119906 119892V)119889 (119892119906 119892V) le 119889 (119892119906 119892V) (38)

which is a contradictionTherefore we conclude that 119892119906 = 119892V If we take 120572(V 119906) ge1 then we also get the same resultSecondly If F and 119892 are weakly compatible then for any119906 isin 119883 if 119906 = 119865V = 119892V (because F and 119892 have a point

of coincidence proven earlier in Step 3) then using weakcompatibility of F and 119892 we get

119865119906 = 119865 (119892V) = 119892 (119865V) = 119892119906 (39)

Thus 119906 is a coincidence point of F and 119892 then using theresult in first part 119892119906 = 119892V which leads to 119865119906 = 119892119906 = 119892V =119865V = 119906

Therefore 119906 is a common fixed point of F and 119892We can prove the uniqueness of the common fixed point

of F and 119892 by making use of condition (19) and assumption(A1) of hypothesis The proof is very simple therefore we donot go through details

If we consider 119892 as identity mapping in the abovetheorem we deduce the following corollary

Corollary 20 Let (119883 119889) be a complete b-metric space and let119865 119883 rarr 119883 be continuous 120572-admissible mapping satisfying thefollowing condition

[120572 (119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120573(119889(119909119910))119889(119909119910) (40)

for all 119909 119910 isin 119883 and 120573 isin 119861119904 where 1 lt 120575If 120572 is transitive mapping and there exists 1199090 isin 119883 such that120572(1199090 1198651199090) ge 1 then F has a fixed point

Taking 120572(119909 119910) = 1 and 120575 = 1 in Corollary 20 we get thefollowing variant of Geraghty theorem

Corollary 21 Let (119883 119889) be a complete b-metric space and let119865 119883 rarr 119883 be self-mapping satisfying the following condition

119889 (119865119909 119865119910) le 120573 (119889 (119909 119910)) 119889 (119909 119910) (41)

for all 119909 119910 isin 119883 and 120573 isin 119861119904 Then F has a unique fixed point119911 isin 119883 and for each119909 isin 119883 the Picard sequence 119865119899119909 convergesto 119911 when 119899 rarr infin

The following lemma derived from [7] is very useful toprove our next theorem

Lemma 22 (see [7]) Let 119910119899 be a sequence in a metric typespace or b-metric space (X d) such that

119889 (119910119899+1 119910119899) le 120582119889 (119910119899 119910119899minus1) (42)

for some 120582 0 lt 120582 lt 1119904 and each 119899 = 1 2 Then 119910119899 is aCauchy sequence in X

Theorem 23 Let (119883 119889) be a complete b-metric space and be120572-regular with respect to 119892 And let 119865 119892 119883 rarr 119883 be two self-mappings where F is 119892-120572-admissible and 120572 is transitive andfor 1199090 isin 119883 one has 120572(1198921199090 1198651199090) ge 1 If 119865(119909) sube 119892(119909) and oneof these two subsets of 119883 is complete suppose that there exists120582 isin [0 1119904) such that

[120572 (119892119909 119892119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119865119892119909119910) (43)

for all 119909 119910 isin 119883 and 1 lt 120575Consider

119872(119865 119892 119909 119910) = max119889 (119892119909 119892119910) 119889 (119892119909 119865119909) 119889 (119892119910 119865119910) 119889 (119892119909 119865119910) + 119889 (119892119910 119865119909)2119904

(44)

Then F and 119892 have a coincidence point

Moreover if F and 119892 are weakly compatible and hypoth-esis has one more additional assumption

(A1) either 120572(119906 V) ge 1 or 120572(V 119906) ge 1 whenever 119865119906 = 119892119906and 119865V = 119892V

then F and 119892 have a unique common fixed point

6 Abstract and Applied Analysis

Proof Let 1199090 isin 119883 be arbitrary and using condition119865(119883) sube 119892(119883) we can easily construct a Jungck sequence 119910119899satisfying

119910119899 = 119865119909119899 = 119892119909119899+1 forall119899 isin N (45)

Now if119910119899 = 119910119899+1 for any 119899 isin N then119892119909119899+1 = 119910119899 = 119910119899+1 =119865119909119899+1 and F and 119892 have a point of coincidenceWithout loss of generality we assume that 119910119899 = 119910119899+1 for

each 119899 isin NHence F is 119892-120572-admissible and 120572(1198921199090 1198921199091) = 120572(11989211990901198651199090) ge 1 Similarly 120572(1198921199091 1198921199092) = 120572(1198651199090 1198651199091) ge 1 By

induction we can easily deduce

120572 (119892119909119899minus1 119892119909119899) ge 1 forall119899 isin N (46)

Step 1 We shall show that 119910119899 is a b-Cauchy sequence Nowtaking (43) and (46) we obtain for each 119899 isin N

120575119889(119910119899+1 119910119899) = 120575119889(119865119909119899+1 119865119909119899)le [120572 (119892119909119899+1 119892119909119899) minus 1 + 120575]119889(119865119909119899+1 119865119909119899)le 120575120582119872(119865119892119909119899+1119909119899)

(47)

This implies that

119889 (119910119899+1 119910119899) le 120582119872(119865 119892 119909119899+1 119909119899) (48)

or

119889 (119910119899+1 119910119899) le 120582max119889 (119892119909119899+1 119892119909119899) 119889 (119892119909119899+1 119865119909119899+1) 119889 (119892119909119899 119865119909119899) 119889 (119892119909119899+1 119865119909119899) + 119889 (119892119909119899 119865119909119899+1)2119904

le 120582max119889 (119910119899 119910119899minus1) 119889 (119910119899 119910119899+1) 119889 (119910119899minus1 119910119899) 119889 (119910119899 119910119899) + 119889 (119910119899minus1 119910119899+1)2119904

le 120582max119889 (119910119899 119910119899minus1) 119889 (119910119899minus1 119910119899) + 119889 (119910119899 119910119899+1)2

(49)

Now if max 119889(119910119899 119910119899minus1) (119889(119910119899minus1 119910119899) + 119889(119910119899 119910119899+1))2 =(119889(119910119899minus1 119910119899)+119889(119910119899 119910119899+1))2 then 119889(119910119899 119910119899minus1) lt (119889(119910119899minus1 119910119899)+119889(119910119899 119910119899+1))2 lt 119889(119910119899+1 119910119899)Then (49) implies that

119889 (119910119899+1 119910119899) le 120582119889 (119910119899+1 119910119899) (50)

which is impossible as 120582 lt 1Therefore we deduce that max 119889(119910119899 119910119899minus1) (119889(119910119899minus1 119910119899)+119889(119910119899 119910119899+1))2 = 119889(119910119899 119910119899minus1)This in turn implies that

119889 (119910119899+1 119910119899) le 120582119889 (119910119899 119910119899minus1) (51)

Using Lemma 22 we obtain that 119910119899 is a b-Cauchysequence and by hypothesis we can suppose that 119892(119883) is tobe complete subspace of 119883 (the proof when 119865(119883) is similar)Then b-completeness of 119892(119883) implies that 119910119899 = 119865119909119899 =119892119909119899+1 b-converges to a point 119906 isin 119892(119883) where 119906 = 119892119911 forsome 119911 isin 119883 or in other words

lim119899rarrinfin

119865119909119899 = lim119899rarrinfin

119892119909119899 = 119892119911 for some 119911 isin 119883 (52)

Step 2 Next we will prove that 119865119911 = 119892119911 Since119883 is 120572-regularwith respect to 119892 and using (52) we have

120572 (119892119909119899(119901) 119892119911) ge 1 forall119901 isin N (53)

To show 119865119911 = 119892119911 we consider120575119889(119865119909119899(119901) 119865119911)le [120572 (119892119909119899(119901) 119892119911) minus 1 + 120575]119889(119865119909119899(119901) 119865119911) le 120575120582119872(119865119892119909119899(119901) 119911)

(54)

This implies that

119889 (119865119909119899(119901) 119865119911) le 120582119872(119865 119892 119909119899(119901) 119911) (55)

or

119889 (119865119909119899(119901) 119865119911) le 120582max119889 (119892119909119899(119901) 119892119911) 119889 (119892119909119899(119901) 119865119909119899(119901)) 119889 (119892119911 119865119911) 119889 (119892119909119899(119901) 119865119911) + 119889 (119892119911 119865119909119899(119901))2119904

(56)

Also119865119909119899(119901) rarr 119892119911 and 119892119909119899(119901) rarr 119892119911 as 119899 rarr infin implyingthat

119889 (119892119909119899(119901) 119865119909119899(119901)) = 119904119889 (119892119909119899(119901) 119892119911) + 119904119889 (119892119911 119865119909119899(119901))997888rarr 0 as 119899 997888rarr infin (57)

Then we have only two cases

Case 1

119889 (119865119909119899(119901) 119865119911) le 120582119889 (119892119911 119865119911)le 120582119904 (119889 (119892119911 119865119909119899(119901)) + 119889 (119865119909119899(119901) 119865119911))

(1 minus 120582119904) 119889 (119865119909119899(119901) 119865119911) le 120582119904119889 (119892119911 119865119909119899(119901)) 997888rarr 0as 119899 997888rarr infin

(58)

Abstract and Applied Analysis 7

Since 1 minus 120582119904 gt 0 it follows that 119865119909119899(119901) rarr 119865119911Case 2

119889 (119865119909119899(119901) 119865119911) le 120582119889 (119892119909119899(119901) 119865119911)2119904le 120582119904(119889 (119892119909119899(119901) 119865119909119899(119901)) + 119889 (119865119909119899(119901) 119865119911)2119904 )

(1 minus 1205822)119889 (119865119909119899(119901) 119865119911) le 1205822119889 (119892119909119899(119901) 119865119909119899(119901)) 997888rarr 0as 119899 997888rarr infin

(59)

Since 1 minus 1205822 gt 0 it follows that 119865119909119899(119901) rarr 119865119911Uniqueness of the limit of a sequence implies that 119865119911 =119892119911Using conditions (A1) and weak compatibility of F and 119892

of hypothesis we can easily prove that F and 119892 have a uniquecommon fixed point

From the above theorem one can deduce the followingcorollary easily

Corollary 24 Let (119883 119889) be a complete b-metric space and 119865 119883 rarr 119883 is continuous and 120572-admissible Suppose that thereexists 120582 isin [0 1119904) such that

[120572 (119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119909119910) (60)

for all 119909 119910 isin 119883 and 1 lt 120575Consider

119872(119909 119910) = max119889 (119909 119910) 119889 (119909 119865119909) 119889 (119910 119865119910) 119889 (119909 119865119910) + 119889 (119910 119865119909)

2119904 (61)

If 120572 is a transitive mapping and there exists 1199090 isin 119883 suchthat 120572(1199090 1198651199090) ge 1 then 119865 has a fixed point

Taking 120572(119909 119910) = 1 and 120575 = 1 we get the following variantof Corollary 312 of [7]

Corollary 25 Let (119883 119889) be a complete b-metric space and let119865 119883 rarr 119883 be self-mapping Suppose that there exists 120582 isin[0 1119904) such that for all 119909 119910 isin 119883119889 (119865119909 119865119910) le 120582119872(119909 119910) (62)

where

119872(119909 119910) = max119889 (119909 119910) 119889 (119909 119865119909) 119889 (119910 119865119910) 119889 (119909 119865119910) + 119889 (119910 119865119909)

2119904 (63)

Then F has a unique fixed point 119911 isin 119883

In what follows we furnish illustrative examples whereinone demonstrates Theorems 19 and 23 on the existence anduniqueness of a common fixed point

Example 26 Let119883 = 0 1 3 be a b-metric space withmetricd given by 119889(119909 119910) = (119909 minus 119910)2 with 119904 = 2 Consider themappings 119865 119892 119883 rarr 119883 defined by 119865(0) = 1 119865(1) = 1and 119865(3) = 1 and 119892 by 119892(0) = 1 119892(1) = 1 and 119892(3) = 3 Letus take 120573 isin 119861119904 as 120573(119905) = 12 for 119905 gt 0 and 120573(0) isin [0 12)

And 120572 119883 times 119883 rarr [0infin) by

120572 (119909 119910) = 1 when 119909 119910 ge 00 otherwise

(64)

then one can examine easily that

119889 (1198650 1198651) = 119889 (1 1) = 0120573 (119889 (1198920 1198921)) 119889 (1198920 1198921) = 120573 (119889 (1 1)) 119889 (1 1) = 0

119889 (1198650 1198653) = 119889 (1 1) = 0120573 (119889 (1198920 1198923)) 119889 (1198920 1198923) = 120573 (119889 (1 3)) 119889 (1 3) = 2

119889 (1198651 1198653) = 119889 (1 1) = 0120573 (119889 (1198921 1198923)) 119889 (1198921 1198923) = 120573 (119889 (1 3)) 119889 (1 3) = 2

(65)

Thus in all cases we get

[120572 (119892119909 119892119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120573(119889(119892119909119892119910))119889(119892119909119892119910) (66)

for all 119909 119910 isin 119883 where 1 lt 120575Also we can check that F and 119892 satisfy all the other

assumptions of Theorem 19 and thus the pair F and 119892 has aunique common fixed point 119909 = 1Example 27 Let119883 = 0 1 3 be a b-metric space with metricd given by 119889(119909 119910) = (119909 minus 119910)2 with 119904 = 2 Consider themappings 119865 119892 119883 rarr 119883 defined by 119865(0) = 0 119865(1) = 1and 119865(3) = 0 and 119892 by 119892(0) = 3 119892(1) = 1 and 119892(3) = 0 Letus take 120582 isin [0 12)

And 120572 119883 times 119883 rarr [0infin) by

120572 (119909 119910) = 1 when 119909 119910 ge 00 otherwise

(67)

Now we verify Theorem 23 by considering the followingcases

8 Abstract and Applied Analysis

Case 1 (take points 0 and 1)

119889 (1198650 1198651) = 119889 (0 1) = 1119889 (1198920 1198921) = 119889 (3 1) = 4119889 (1198920 1198650) = 119889 (3 0) = 9119889 (1198921 1198651) = 119889 (1 1) = 0119889 (1198920 1198651) + 119889 (1198921 1198650)

2 sdot 2 = 119889 (3 1) + 119889 (1 0)4 = 4 + 14= 54

119872 (119865 119892 0 1) = max119889 (1198920 1198921) 119889 (1198920 1198650) 119889 (1198921 1198651) 119889 (1198920 1198651) + 119889 (1198921 1198650)4 = 4 9 0 54= 9

(68)

Case 2 (take points 1 and 3)

119889 (1198651 1198653) = 119889 (1 1) = 0119872 (119865 119892 1 3) = max119889 (1198921 1198923) 119889 (1198921 1198651) 119889 (1198923 1198653) 119889 (1198921 1198653) + 119889 (1198923 1198651)4 = 1 0 0 24= 1

(69)

Case 3 (take points 0 and 3)

119889 (1198650 1198653) = 119889 (0 0) = 0119872 (119865 119892 0 3) = max119889 (1198920 1198923) 119889 (1198920 1198650) 119889 (1198923 1198653) 119889 (1198920 1198653) + 119889 (1198923 1198650)4 = 9 9 0 94= 9

(70)

Thus in all cases F and 119892 satisfy all the assumptions ofTheorem 23 and thus the pair F and 119892 has a unique commonfixed point 119909 = 13 An Application to an Integral Equation

This section deals with the applications of results proven inthe previous section Here we will investigate the solution ofintegral equation through our results

Consider the following integral equation

119909 (119905) = int10119870 (119905 119903 119909 (119903)) 119889119903 (71)

where119870 [0 1] times [0 1] timesR rarr R

Let 119883 = 119862[0 1] be the set of continuous real functionsdefined on [0 1] Define the b-metric by 119889(119909(119905) 119910(119905)) =max119905isin[01](|119909(119905)| + |119910(119905)|)119901 for all 119909 119910 isin 119883

Consider 119901 gt 1 Then (119883 119889) is a complete b-metric spacewith the constant 119904 = 2119901minus1119865(119909(119905)) = int1

0119870(119905 119903 119909(119903))119889119903 for all 119909 isin 119883 and for all 119905 isin[0 1] Then the existence of a solution to (71) is equivalent to

the existence of a fixed point of F Nowwe prove the followingresult

Theorem 28 Let us suppose that the following hypotheseshold

(i) 119870 [0 1] times [0 1] timesR rarr R is continuous

(ii) For all 119905 119903 isin [0 1] there exists a continuous operator120585 [0 1] times [0 1] rarr R such that

|119870 (119905 119903 119909 (119903))| + 1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816lt 1205821119901120585 (119905 119903) (|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)

sup119905isin[01]

int10120585 (119905 119903) 119889119903 le 1

(72)

where 0 lt 120582 lt 1119904(iii) There exists a function 120583 R2 rarr R such that for all119905 isin 119868 and for all 119886 119887 isin R with 120583(119886 119887) ge 0 we have

120583(1199091 (119905) int10119870 (119905 119903 119909 (119903)) 119889119903) ge 0

for 119909 isin 119862 (119868) forall119905 isin 119868120583 (119909 (119905) 119910 (119905)) ge 0

120583 (int10119870 (119905 119903 119909 (119903)) int1

0119870(119905 119903 119910 (119903)) 119889119903) ge 0

forall119905 isin 119868 forall119909 119910 isin 119862 (119868)

(73)

Then the integral equation (71) has a unique solution 119909 isin119883Proof We define 120572 119862(119868) times 119862(119868) rarr [0infin) by

120572 (119909 119910) = 1 when 120583 (119909 (119905) 119910 (119905)) ge 00 otherwise

(74)

Then for all 119909 119910 isin 119862(119868) we have 120572(119909 119910) = 1 and120572(119910 119911) = 1 implying that 120572(119909 119911) = 1 for all 119909 119910 119911 isin 119862(119868)which proves that 120572 is a transitive mapping

If 120572(119909 119910) = 1 for all 119909 119910 isin 119862(119868) then 120583(119909(119905) 119910(119905)) ge 0From (iii) we have 120583(119865119909(119905) 119865119910(119905)) ge 0 and so 120572(119865119909 119865119910) = 1Thus F is 120572-admissible

From (iii) there exists 1199091 isin 119862(119868) such that 120572(1199091 1198651199091) = 1

Abstract and Applied Analysis 9

Also we observe from (72) that

(|119865119909 (119905)| + 1003816100381610038161003816119865119910 (119905)1003816100381610038161003816)119901= (100381610038161003816100381610038161003816100381610038161003816int

1

0119870 (119905 119903 119909 (119903)) 119889119903100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816int1

0119870(119905 119903 119910 (119903)) 119889119903100381610038161003816100381610038161003816100381610038161003816)

119901

le (int10|119870 (119905 119903 119909 (119903))| 119889119903 + int1

0

1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816 119889119903)119901

le (int10(|119870 (119905 119903 119909 (119903))| + 1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816) 119889119903)

119901

le (int10(1205821119901120585 (119905 119903) (|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)) 119889119903)

119901

= (int10(1205821119901120585 (119905 119903) ((|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)119901)1119901)119889119903)

119901

le (int101205821119901120585 (119905 119903) 1198891119901 (119909 (119903) 119910 (119903)) 119889119903)119901

= 120582119889 (119909 (119905) 119910 (119905)) (int10120585 (119905 119903) 119889119903)119901

le 120582119889 (119909 (119905) 119910 (119905)) le 120582119872(119909 (119905) 119910 (119905))

(75)

which in turn implies that [120572(119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119909119910)Now all the conditions of Corollary 24 hold and F has a

unique fixed point 119909 isin 119883 which means that 119909 is the uniquesolution for the integral equation (71)

4 Conclusion

To conclude we can assert that our results are novelinteresting and generalized while considering the alpha-admissible Geraghty type mappings These results extendimprove unify and generalize many theorems based onalpha-admissible mappings in 119887-metric spaces Examples arepresented in a simplest form and illustrate the theoremsApplication to integral equation adds value to our research

Conflicts of Interest

All the authors of this article declare that they have noconflicts of interest regarding the publication of this article

Authorsrsquo Contributions

All authors have equal contribution in writing this paper Allauthors read and approved the final paper

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] R Allahyari R Arab and A S Haghighi ldquoFixed pointsof admissible almost contractive type mappings on b-metric

spaces with an application to quadratic integral equationsrdquoJournal of Inequalities and Applications vol 2015 no 32 18pages 2015

[3] M-F Bota E Karapinar andOMlesnite ldquoUlam-hyers stabilityresults for fixed point problems via 120572-120595-contractive mappingin (b)-metric spacerdquo Abstract and Applied Analysis vol 2013Article ID 825293 6 pages 2013

[4] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacerdquo Functional Analysis vol 30 pp 26ndash37 1989

[5] C Chen J Dong and C Zhu ldquoSome fixed point theorems inb-metric-like spacesrdquo Fixed Point Theory and Applications vol2015 no 122 10 pages 2015

[6] H-SDingM Imdad S Radenovic and J Vujakovic ldquoOn somefixed point results in b-metric rectangular and b-rectangularmetric spacesrdquo Arab Journal of Mathematical Sciences vol 22(2016) pp 151ndash164 2015

[7] S Radenovic M Jovanovic and Z Kadelburg ldquoCommon fixedpoint results in metric-type spacesrdquo Fixed Point Theory andApplications vol 2010 article 978121 15 pages 2010

[8] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 article 315398 2010

[9] J R Roshan V Parvaneh and Z Kadelburg ldquoCommon fixedpoint theorems for weakly isotone increasing mappings inordered b-metric spacesrdquo Journal of Nonlinear Science and itsApplications JNSA vol 7 no 4 pp 229ndash245 2014

[10] W Sintunavarat S Plubtieng and P Katchang ldquoFixed pointresult and applications on a b-metric space endowed with anarbitrary binary relationrdquo Fixed Point Theory and Applicationsvol 2013 article 296 2013

[11] W Sintunavarat ldquoGeneralized Ulam-Hyers stability well-posedness and limit shadowing of fixed point problems for 120572-120573-contraction mapping in metric spacesrdquo The Scientific WorldJournal vol 2014 Article ID 569174 7 pages 2014

[12] R J Shahkoohi and A Razani ldquoSome fixed point theoremsfor rational Geraghty contractivemappings in ordered b-metricspacesrdquo Journal of Inequalities and Applications vol 373 p 232014

[13] S L Singh and B Prasad ldquoSome coincidence theorems andstability of iterative proceduresrdquoComputersampMathematics withApplications vol 55 no 11 pp 2512ndash2520 2008

[14] M A Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[15] S Cho J Bae and E Karapınar ldquoFixed point theorems for 120572-Geraghty contraction type maps in metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 329 11 pages 2013

[16] D Dukic Z Kadelburg and S Radenovic ldquoFixed points ofGeraghty-type mappings in various generalized metric spacesrdquoAbstract and Applied Analysis vol 2011 Article ID 561245 13pages 2011

[17] M E Gordji M Ramezani Y J Cho and S Pirbavafa ldquoAgeneralization ofGeraghtyrsquos theorem in partially orderedmetricspaces and applications to ordinary differential equationsrdquoFixed Point Theory and Applications vol 2012 article 74 2012

[18] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

10 Abstract and Applied Analysis

[19] A Felhi S Sahmim and H Aydi ldquoUlam-hyers stability andwell-posedness of fixed point problems for 120572-120582-metric spacesrdquoFixed Point Theory and Applications vol 2016 no 1 20 pages2016

[20] V L Rosa and P Vetro ldquoCommon fixed points for 120572-120595-120601-contractions in generalized metric spacesrdquo Nonlinear AnalysisModelling and Control vol 19 no 1 pp 43ndash54 2014

Submit your manuscripts athttpswwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Proof Let1199090 isin 119883 such that120572(1198921199090 1198651199090) ge 1 In order to provethat F and 119892 have a point of coincidence and using 119865(119909) sube119892(119909) define two sequences 119909119899 and 119910119899 in X such that

119910119899 = 119865119909119899 = 119892119909119899+1 forall119899 isin N (20)

Now if119910119899 = 119910119899+1 for any 119899 isin N then119892119909119899+1 = 119910119899 = 119910119899+1 =119865119909119899+1 and F and 119892 have a point of coincidenceWithout loss of generality one can suppose that119910119899 = 119910119899+1

for each 119899 isin NHence 119865 is 119892-120572-admissible and 120572(1198921199090 1198921199091) =120572(1198921199090 1198651199090) ge 1 Similarly 120572(1198921199091 1198921199092) = 120572(1198651199090 1198651199091) ge 1

By induction we can easily deduce

120572 (119892119909119899minus1 119892119909119899) ge 1 forall119899 isin N (21)

Step 1 We shall show that lim119899rarrinfin119889(119910119899+1 119910119899) = 0 Now itfollows from (19) and (21) for each 119899 isin N that

120575119889(119910119899+1 119910119899) = 120575119889(119865119909119899+1 119865119909119899)le [120572 (119892119909119899+1 119892119909119899) minus 1 + 120575]119889(119865119909119899+1 119865119909119899)le 120575120573(119889(119892119909119899+1 119892119909119899))119889(119892119909119899+1 119892119909119899)

(22)

This implies that

119889 (119910119899+1 119910119899) le 120573 (119889 (119892119909119899+1 119892119909119899)) 119889 (119892119909119899+1 119892119909119899)le 1119904 119889 (119910119899 119910119899minus1) forall119899 isin N (23)

Further from (23) it follows that

lim119899rarrinfin

119889 (119910119899+1 119910119899) = 0 (24)

Step 2 Next we show that 119910119899 is a Cauchy sequence in b-metric space 119883 On the contrary assume that 119910119899 is not aCauchy sequence Then there exists 120576 gt 0 and subsequencesof integers 119899119901 and119898119901 with 119899119901 gt 119898119901 ge 0 such that

119889 (119910119898119901 119910119899119901) gt 120576119889 (119910119898119901 119910119899119901minus1) lt 120576

forall119901 isin N(25)

Since 119899119901 gt 119898119901 and120572 is transitivemapping we can deduceeasily by using triangle inequality

120572 (119892119909119898119901+1 119892119909119899119901) ge 1 forall119901 isin N (26)

Now using (19) and (26) we can consider

120575119889(119910119898119901+1 119910119899119901 ) = 120575119889(119865119909119898119901+1 119865119909119899119901 )le [120572 (119892119909119898119901+1 119892119909119899119901) minus 1 + 120575]119889(119865119909119898119901+1 119865119909119899119901 )le 120575120573(119889(119892119909119898119901+1 119892119909119899119901 ))119889(119892119909119898119901+1 119892119909119899119901 )

(27)

This gives

119889 (119910119898119901+1 119910119899119901)le 120573 (119889 (119892119909119898119901+1 119892119909119899119901)) 119889 (119892119909119898119901+1 119892119909119899119901)le 120573 (119889 (119910119898119901 119910119899119901minus1)) 119889 (119910119898119901 119910119899119901minus1)le 120573 (119889 (119910119898119901 119910119899119901minus1)) 120576

(28)

119889 (119910119898119901+1 119910119899119901)120576 le 120573 (119889 (119910119898119901 119910119899119901minus1)) lt 1119904 (29)

with 119899119901 gt 119898119901 gt 119901 for all 119901 isin NHence by Lemma 4 and (29) we have

1119904 = 1120576 120576119904 le lim119901rarrinfin

inf 1120576 119889 (119910119898119901+1 119910119899119901)le lim119901rarrinfin

sup 1120576 119889 (119910119898119901+1 119910119899119901)le lim119901rarrinfin

inf 120573 (119889 (119910119898119901 119910119899119901minus1))le lim119901rarrinfin

sup120573 (119889 (119910119898119901 119910119899119901minus1)) le 1119904

(30)

that is

lim119901rarrinfin

120573 (119889 (119910119898119901 119910119899119901minus1)) = 1119904 (31)

Also 120573 isin 119861119904 implies that lim119901rarrinfin119889(119910119898119901 119910119899119901minus1) = 0However this is not possible as using (4) of Lemma 4 weget that

120576119904 le 1119904 119889 (119910119898119901 119910119899119901) le 119889 (119910119898119901 119910119899119901minus1) + 119889 (119910119899119901minus1 119910119899119901)997888rarr 0

(32)

as 119901 rarr infin which leads to contradicting (25)Therefore 119910119899 is a b-Cauchy sequence and by hypothesis

we can suppose that 119892(119883) is to be complete subspace ofX (proof can be derived in a similar manner when 119865(119883)is supposed to be complete) Then b-completeness of 119892(119883)implies that 119910119899 = 119865119909119899 = 119892119909119899+1 b-converges to a point119906 isin 119892(119883) where 119906 = 119892119911 for some 119911 isin 119883 or in other words

lim119899rarrinfin

119865119909119899 = lim119899rarrinfin

119892119909119899 = 119892119911 for some 119911 isin 119883 (33)

Step 3 Next we will prove that 119865119911 = 119892119911 Since X is 120572-regularwith respect to 119892 and using (33) we have

120572 (119892119909119899(119901) 119892119911) ge 1 forall119901 isin N (34)

Abstract and Applied Analysis 5

To show 119865119911 = 119892119911 we consider120575119889(119865119911119892119911) le 120575119904119889(119865119911119865119909119899(119901))+119904119889(119892119909119899(119901)+1 119892119911)

le 120575119904119889(119865119911119865119909119899(119901)) times 120575119904119889(119892119909119899(119901)+1119892119911)le [120572 (119892119911 119892119909119899(119901)) minus 1 + 120575]119904119889(119865119911119865119909119899(119901))times 120575119904119889(119892119909119899(119901)+1119892119911)

le 120575119904120573(119889(119892119911119892119909119899(119901)))119889(119892119911119892119909119899(119901)) times 120575119904119889(119892119909119899(119901)+1 119892119911)

(35)

This in turns implies that

119889 (119865119911 119892119911) le 119904120573 (119889 (119892119911 119892119909119899(119901))) 119889 (119892119911 119892119909119899(119901))+ 119904119889 (119892119909119899(119901)+1 119892119911)

1119904 119889 (119865119911 119892119911) le 120573 (119889 (119892119911 119892119909119899(119901))) 119889 (119892119911 119892119909119899(119901))+ 119889 (119892119909119899(119901)+1 119892119911)

lt 1119904 119889 (119892119911 119892119909119899(119901)) + 119889 (119892119909119899(119901)+1 119892119911)997888rarr 0 as 119899 997888rarr infin

(36)

Thus 119865119911 = 119892119911 = 119906 is a point of coincidence of F and 119892Step 4 Next we prove that F and 119892 have a common fixedpoint Firstly we claim that if 119865119906 = 119892119906 and 119865V = 119892V then119892119906 = 119892V By hypotheses 120572(119906 V) ge 1 or 120572(V 119906) ge 1 Supposethat 120572(119906 V) ge 1 and consider that

120575119889(119892119906119892V) = 120575119889(119865119906119865V) le [120572 (119892119906 119892V) minus 1 + 120575]119889(119865119906119865V)le 120575120573(119889(119892119906119892V))119889(119892119906119892V) (37)

Then we get that

119889 (119892119906 119892V) le 120573 (119889 (119892119906 119892V)) 119889 (119892119906 119892V)119889 (119892119906 119892V) le 119889 (119892119906 119892V) (38)

which is a contradictionTherefore we conclude that 119892119906 = 119892V If we take 120572(V 119906) ge1 then we also get the same resultSecondly If F and 119892 are weakly compatible then for any119906 isin 119883 if 119906 = 119865V = 119892V (because F and 119892 have a point

of coincidence proven earlier in Step 3) then using weakcompatibility of F and 119892 we get

119865119906 = 119865 (119892V) = 119892 (119865V) = 119892119906 (39)

Thus 119906 is a coincidence point of F and 119892 then using theresult in first part 119892119906 = 119892V which leads to 119865119906 = 119892119906 = 119892V =119865V = 119906

Therefore 119906 is a common fixed point of F and 119892We can prove the uniqueness of the common fixed point

of F and 119892 by making use of condition (19) and assumption(A1) of hypothesis The proof is very simple therefore we donot go through details

If we consider 119892 as identity mapping in the abovetheorem we deduce the following corollary

Corollary 20 Let (119883 119889) be a complete b-metric space and let119865 119883 rarr 119883 be continuous 120572-admissible mapping satisfying thefollowing condition

[120572 (119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120573(119889(119909119910))119889(119909119910) (40)

for all 119909 119910 isin 119883 and 120573 isin 119861119904 where 1 lt 120575If 120572 is transitive mapping and there exists 1199090 isin 119883 such that120572(1199090 1198651199090) ge 1 then F has a fixed point

Taking 120572(119909 119910) = 1 and 120575 = 1 in Corollary 20 we get thefollowing variant of Geraghty theorem

Corollary 21 Let (119883 119889) be a complete b-metric space and let119865 119883 rarr 119883 be self-mapping satisfying the following condition

119889 (119865119909 119865119910) le 120573 (119889 (119909 119910)) 119889 (119909 119910) (41)

for all 119909 119910 isin 119883 and 120573 isin 119861119904 Then F has a unique fixed point119911 isin 119883 and for each119909 isin 119883 the Picard sequence 119865119899119909 convergesto 119911 when 119899 rarr infin

The following lemma derived from [7] is very useful toprove our next theorem

Lemma 22 (see [7]) Let 119910119899 be a sequence in a metric typespace or b-metric space (X d) such that

119889 (119910119899+1 119910119899) le 120582119889 (119910119899 119910119899minus1) (42)

for some 120582 0 lt 120582 lt 1119904 and each 119899 = 1 2 Then 119910119899 is aCauchy sequence in X

Theorem 23 Let (119883 119889) be a complete b-metric space and be120572-regular with respect to 119892 And let 119865 119892 119883 rarr 119883 be two self-mappings where F is 119892-120572-admissible and 120572 is transitive andfor 1199090 isin 119883 one has 120572(1198921199090 1198651199090) ge 1 If 119865(119909) sube 119892(119909) and oneof these two subsets of 119883 is complete suppose that there exists120582 isin [0 1119904) such that

[120572 (119892119909 119892119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119865119892119909119910) (43)

for all 119909 119910 isin 119883 and 1 lt 120575Consider

119872(119865 119892 119909 119910) = max119889 (119892119909 119892119910) 119889 (119892119909 119865119909) 119889 (119892119910 119865119910) 119889 (119892119909 119865119910) + 119889 (119892119910 119865119909)2119904

(44)

Then F and 119892 have a coincidence point

Moreover if F and 119892 are weakly compatible and hypoth-esis has one more additional assumption

(A1) either 120572(119906 V) ge 1 or 120572(V 119906) ge 1 whenever 119865119906 = 119892119906and 119865V = 119892V

then F and 119892 have a unique common fixed point

6 Abstract and Applied Analysis

Proof Let 1199090 isin 119883 be arbitrary and using condition119865(119883) sube 119892(119883) we can easily construct a Jungck sequence 119910119899satisfying

119910119899 = 119865119909119899 = 119892119909119899+1 forall119899 isin N (45)

Now if119910119899 = 119910119899+1 for any 119899 isin N then119892119909119899+1 = 119910119899 = 119910119899+1 =119865119909119899+1 and F and 119892 have a point of coincidenceWithout loss of generality we assume that 119910119899 = 119910119899+1 for

each 119899 isin NHence F is 119892-120572-admissible and 120572(1198921199090 1198921199091) = 120572(11989211990901198651199090) ge 1 Similarly 120572(1198921199091 1198921199092) = 120572(1198651199090 1198651199091) ge 1 By

induction we can easily deduce

120572 (119892119909119899minus1 119892119909119899) ge 1 forall119899 isin N (46)

Step 1 We shall show that 119910119899 is a b-Cauchy sequence Nowtaking (43) and (46) we obtain for each 119899 isin N

120575119889(119910119899+1 119910119899) = 120575119889(119865119909119899+1 119865119909119899)le [120572 (119892119909119899+1 119892119909119899) minus 1 + 120575]119889(119865119909119899+1 119865119909119899)le 120575120582119872(119865119892119909119899+1119909119899)

(47)

This implies that

119889 (119910119899+1 119910119899) le 120582119872(119865 119892 119909119899+1 119909119899) (48)

or

119889 (119910119899+1 119910119899) le 120582max119889 (119892119909119899+1 119892119909119899) 119889 (119892119909119899+1 119865119909119899+1) 119889 (119892119909119899 119865119909119899) 119889 (119892119909119899+1 119865119909119899) + 119889 (119892119909119899 119865119909119899+1)2119904

le 120582max119889 (119910119899 119910119899minus1) 119889 (119910119899 119910119899+1) 119889 (119910119899minus1 119910119899) 119889 (119910119899 119910119899) + 119889 (119910119899minus1 119910119899+1)2119904

le 120582max119889 (119910119899 119910119899minus1) 119889 (119910119899minus1 119910119899) + 119889 (119910119899 119910119899+1)2

(49)

Now if max 119889(119910119899 119910119899minus1) (119889(119910119899minus1 119910119899) + 119889(119910119899 119910119899+1))2 =(119889(119910119899minus1 119910119899)+119889(119910119899 119910119899+1))2 then 119889(119910119899 119910119899minus1) lt (119889(119910119899minus1 119910119899)+119889(119910119899 119910119899+1))2 lt 119889(119910119899+1 119910119899)Then (49) implies that

119889 (119910119899+1 119910119899) le 120582119889 (119910119899+1 119910119899) (50)

which is impossible as 120582 lt 1Therefore we deduce that max 119889(119910119899 119910119899minus1) (119889(119910119899minus1 119910119899)+119889(119910119899 119910119899+1))2 = 119889(119910119899 119910119899minus1)This in turn implies that

119889 (119910119899+1 119910119899) le 120582119889 (119910119899 119910119899minus1) (51)

Using Lemma 22 we obtain that 119910119899 is a b-Cauchysequence and by hypothesis we can suppose that 119892(119883) is tobe complete subspace of 119883 (the proof when 119865(119883) is similar)Then b-completeness of 119892(119883) implies that 119910119899 = 119865119909119899 =119892119909119899+1 b-converges to a point 119906 isin 119892(119883) where 119906 = 119892119911 forsome 119911 isin 119883 or in other words

lim119899rarrinfin

119865119909119899 = lim119899rarrinfin

119892119909119899 = 119892119911 for some 119911 isin 119883 (52)

Step 2 Next we will prove that 119865119911 = 119892119911 Since119883 is 120572-regularwith respect to 119892 and using (52) we have

120572 (119892119909119899(119901) 119892119911) ge 1 forall119901 isin N (53)

To show 119865119911 = 119892119911 we consider120575119889(119865119909119899(119901) 119865119911)le [120572 (119892119909119899(119901) 119892119911) minus 1 + 120575]119889(119865119909119899(119901) 119865119911) le 120575120582119872(119865119892119909119899(119901) 119911)

(54)

This implies that

119889 (119865119909119899(119901) 119865119911) le 120582119872(119865 119892 119909119899(119901) 119911) (55)

or

119889 (119865119909119899(119901) 119865119911) le 120582max119889 (119892119909119899(119901) 119892119911) 119889 (119892119909119899(119901) 119865119909119899(119901)) 119889 (119892119911 119865119911) 119889 (119892119909119899(119901) 119865119911) + 119889 (119892119911 119865119909119899(119901))2119904

(56)

Also119865119909119899(119901) rarr 119892119911 and 119892119909119899(119901) rarr 119892119911 as 119899 rarr infin implyingthat

119889 (119892119909119899(119901) 119865119909119899(119901)) = 119904119889 (119892119909119899(119901) 119892119911) + 119904119889 (119892119911 119865119909119899(119901))997888rarr 0 as 119899 997888rarr infin (57)

Then we have only two cases

Case 1

119889 (119865119909119899(119901) 119865119911) le 120582119889 (119892119911 119865119911)le 120582119904 (119889 (119892119911 119865119909119899(119901)) + 119889 (119865119909119899(119901) 119865119911))

(1 minus 120582119904) 119889 (119865119909119899(119901) 119865119911) le 120582119904119889 (119892119911 119865119909119899(119901)) 997888rarr 0as 119899 997888rarr infin

(58)

Abstract and Applied Analysis 7

Since 1 minus 120582119904 gt 0 it follows that 119865119909119899(119901) rarr 119865119911Case 2

119889 (119865119909119899(119901) 119865119911) le 120582119889 (119892119909119899(119901) 119865119911)2119904le 120582119904(119889 (119892119909119899(119901) 119865119909119899(119901)) + 119889 (119865119909119899(119901) 119865119911)2119904 )

(1 minus 1205822)119889 (119865119909119899(119901) 119865119911) le 1205822119889 (119892119909119899(119901) 119865119909119899(119901)) 997888rarr 0as 119899 997888rarr infin

(59)

Since 1 minus 1205822 gt 0 it follows that 119865119909119899(119901) rarr 119865119911Uniqueness of the limit of a sequence implies that 119865119911 =119892119911Using conditions (A1) and weak compatibility of F and 119892

of hypothesis we can easily prove that F and 119892 have a uniquecommon fixed point

From the above theorem one can deduce the followingcorollary easily

Corollary 24 Let (119883 119889) be a complete b-metric space and 119865 119883 rarr 119883 is continuous and 120572-admissible Suppose that thereexists 120582 isin [0 1119904) such that

[120572 (119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119909119910) (60)

for all 119909 119910 isin 119883 and 1 lt 120575Consider

119872(119909 119910) = max119889 (119909 119910) 119889 (119909 119865119909) 119889 (119910 119865119910) 119889 (119909 119865119910) + 119889 (119910 119865119909)

2119904 (61)

If 120572 is a transitive mapping and there exists 1199090 isin 119883 suchthat 120572(1199090 1198651199090) ge 1 then 119865 has a fixed point

Taking 120572(119909 119910) = 1 and 120575 = 1 we get the following variantof Corollary 312 of [7]

Corollary 25 Let (119883 119889) be a complete b-metric space and let119865 119883 rarr 119883 be self-mapping Suppose that there exists 120582 isin[0 1119904) such that for all 119909 119910 isin 119883119889 (119865119909 119865119910) le 120582119872(119909 119910) (62)

where

119872(119909 119910) = max119889 (119909 119910) 119889 (119909 119865119909) 119889 (119910 119865119910) 119889 (119909 119865119910) + 119889 (119910 119865119909)

2119904 (63)

Then F has a unique fixed point 119911 isin 119883

In what follows we furnish illustrative examples whereinone demonstrates Theorems 19 and 23 on the existence anduniqueness of a common fixed point

Example 26 Let119883 = 0 1 3 be a b-metric space withmetricd given by 119889(119909 119910) = (119909 minus 119910)2 with 119904 = 2 Consider themappings 119865 119892 119883 rarr 119883 defined by 119865(0) = 1 119865(1) = 1and 119865(3) = 1 and 119892 by 119892(0) = 1 119892(1) = 1 and 119892(3) = 3 Letus take 120573 isin 119861119904 as 120573(119905) = 12 for 119905 gt 0 and 120573(0) isin [0 12)

And 120572 119883 times 119883 rarr [0infin) by

120572 (119909 119910) = 1 when 119909 119910 ge 00 otherwise

(64)

then one can examine easily that

119889 (1198650 1198651) = 119889 (1 1) = 0120573 (119889 (1198920 1198921)) 119889 (1198920 1198921) = 120573 (119889 (1 1)) 119889 (1 1) = 0

119889 (1198650 1198653) = 119889 (1 1) = 0120573 (119889 (1198920 1198923)) 119889 (1198920 1198923) = 120573 (119889 (1 3)) 119889 (1 3) = 2

119889 (1198651 1198653) = 119889 (1 1) = 0120573 (119889 (1198921 1198923)) 119889 (1198921 1198923) = 120573 (119889 (1 3)) 119889 (1 3) = 2

(65)

Thus in all cases we get

[120572 (119892119909 119892119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120573(119889(119892119909119892119910))119889(119892119909119892119910) (66)

for all 119909 119910 isin 119883 where 1 lt 120575Also we can check that F and 119892 satisfy all the other

assumptions of Theorem 19 and thus the pair F and 119892 has aunique common fixed point 119909 = 1Example 27 Let119883 = 0 1 3 be a b-metric space with metricd given by 119889(119909 119910) = (119909 minus 119910)2 with 119904 = 2 Consider themappings 119865 119892 119883 rarr 119883 defined by 119865(0) = 0 119865(1) = 1and 119865(3) = 0 and 119892 by 119892(0) = 3 119892(1) = 1 and 119892(3) = 0 Letus take 120582 isin [0 12)

And 120572 119883 times 119883 rarr [0infin) by

120572 (119909 119910) = 1 when 119909 119910 ge 00 otherwise

(67)

Now we verify Theorem 23 by considering the followingcases

8 Abstract and Applied Analysis

Case 1 (take points 0 and 1)

119889 (1198650 1198651) = 119889 (0 1) = 1119889 (1198920 1198921) = 119889 (3 1) = 4119889 (1198920 1198650) = 119889 (3 0) = 9119889 (1198921 1198651) = 119889 (1 1) = 0119889 (1198920 1198651) + 119889 (1198921 1198650)

2 sdot 2 = 119889 (3 1) + 119889 (1 0)4 = 4 + 14= 54

119872 (119865 119892 0 1) = max119889 (1198920 1198921) 119889 (1198920 1198650) 119889 (1198921 1198651) 119889 (1198920 1198651) + 119889 (1198921 1198650)4 = 4 9 0 54= 9

(68)

Case 2 (take points 1 and 3)

119889 (1198651 1198653) = 119889 (1 1) = 0119872 (119865 119892 1 3) = max119889 (1198921 1198923) 119889 (1198921 1198651) 119889 (1198923 1198653) 119889 (1198921 1198653) + 119889 (1198923 1198651)4 = 1 0 0 24= 1

(69)

Case 3 (take points 0 and 3)

119889 (1198650 1198653) = 119889 (0 0) = 0119872 (119865 119892 0 3) = max119889 (1198920 1198923) 119889 (1198920 1198650) 119889 (1198923 1198653) 119889 (1198920 1198653) + 119889 (1198923 1198650)4 = 9 9 0 94= 9

(70)

Thus in all cases F and 119892 satisfy all the assumptions ofTheorem 23 and thus the pair F and 119892 has a unique commonfixed point 119909 = 13 An Application to an Integral Equation

This section deals with the applications of results proven inthe previous section Here we will investigate the solution ofintegral equation through our results

Consider the following integral equation

119909 (119905) = int10119870 (119905 119903 119909 (119903)) 119889119903 (71)

where119870 [0 1] times [0 1] timesR rarr R

Let 119883 = 119862[0 1] be the set of continuous real functionsdefined on [0 1] Define the b-metric by 119889(119909(119905) 119910(119905)) =max119905isin[01](|119909(119905)| + |119910(119905)|)119901 for all 119909 119910 isin 119883

Consider 119901 gt 1 Then (119883 119889) is a complete b-metric spacewith the constant 119904 = 2119901minus1119865(119909(119905)) = int1

0119870(119905 119903 119909(119903))119889119903 for all 119909 isin 119883 and for all 119905 isin[0 1] Then the existence of a solution to (71) is equivalent to

the existence of a fixed point of F Nowwe prove the followingresult

Theorem 28 Let us suppose that the following hypotheseshold

(i) 119870 [0 1] times [0 1] timesR rarr R is continuous

(ii) For all 119905 119903 isin [0 1] there exists a continuous operator120585 [0 1] times [0 1] rarr R such that

|119870 (119905 119903 119909 (119903))| + 1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816lt 1205821119901120585 (119905 119903) (|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)

sup119905isin[01]

int10120585 (119905 119903) 119889119903 le 1

(72)

where 0 lt 120582 lt 1119904(iii) There exists a function 120583 R2 rarr R such that for all119905 isin 119868 and for all 119886 119887 isin R with 120583(119886 119887) ge 0 we have

120583(1199091 (119905) int10119870 (119905 119903 119909 (119903)) 119889119903) ge 0

for 119909 isin 119862 (119868) forall119905 isin 119868120583 (119909 (119905) 119910 (119905)) ge 0

120583 (int10119870 (119905 119903 119909 (119903)) int1

0119870(119905 119903 119910 (119903)) 119889119903) ge 0

forall119905 isin 119868 forall119909 119910 isin 119862 (119868)

(73)

Then the integral equation (71) has a unique solution 119909 isin119883Proof We define 120572 119862(119868) times 119862(119868) rarr [0infin) by

120572 (119909 119910) = 1 when 120583 (119909 (119905) 119910 (119905)) ge 00 otherwise

(74)

Then for all 119909 119910 isin 119862(119868) we have 120572(119909 119910) = 1 and120572(119910 119911) = 1 implying that 120572(119909 119911) = 1 for all 119909 119910 119911 isin 119862(119868)which proves that 120572 is a transitive mapping

If 120572(119909 119910) = 1 for all 119909 119910 isin 119862(119868) then 120583(119909(119905) 119910(119905)) ge 0From (iii) we have 120583(119865119909(119905) 119865119910(119905)) ge 0 and so 120572(119865119909 119865119910) = 1Thus F is 120572-admissible

From (iii) there exists 1199091 isin 119862(119868) such that 120572(1199091 1198651199091) = 1

Abstract and Applied Analysis 9

Also we observe from (72) that

(|119865119909 (119905)| + 1003816100381610038161003816119865119910 (119905)1003816100381610038161003816)119901= (100381610038161003816100381610038161003816100381610038161003816int

1

0119870 (119905 119903 119909 (119903)) 119889119903100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816int1

0119870(119905 119903 119910 (119903)) 119889119903100381610038161003816100381610038161003816100381610038161003816)

119901

le (int10|119870 (119905 119903 119909 (119903))| 119889119903 + int1

0

1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816 119889119903)119901

le (int10(|119870 (119905 119903 119909 (119903))| + 1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816) 119889119903)

119901

le (int10(1205821119901120585 (119905 119903) (|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)) 119889119903)

119901

= (int10(1205821119901120585 (119905 119903) ((|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)119901)1119901)119889119903)

119901

le (int101205821119901120585 (119905 119903) 1198891119901 (119909 (119903) 119910 (119903)) 119889119903)119901

= 120582119889 (119909 (119905) 119910 (119905)) (int10120585 (119905 119903) 119889119903)119901

le 120582119889 (119909 (119905) 119910 (119905)) le 120582119872(119909 (119905) 119910 (119905))

(75)

which in turn implies that [120572(119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119909119910)Now all the conditions of Corollary 24 hold and F has a

unique fixed point 119909 isin 119883 which means that 119909 is the uniquesolution for the integral equation (71)

4 Conclusion

To conclude we can assert that our results are novelinteresting and generalized while considering the alpha-admissible Geraghty type mappings These results extendimprove unify and generalize many theorems based onalpha-admissible mappings in 119887-metric spaces Examples arepresented in a simplest form and illustrate the theoremsApplication to integral equation adds value to our research

Conflicts of Interest

All the authors of this article declare that they have noconflicts of interest regarding the publication of this article

Authorsrsquo Contributions

All authors have equal contribution in writing this paper Allauthors read and approved the final paper

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] R Allahyari R Arab and A S Haghighi ldquoFixed pointsof admissible almost contractive type mappings on b-metric

spaces with an application to quadratic integral equationsrdquoJournal of Inequalities and Applications vol 2015 no 32 18pages 2015

[3] M-F Bota E Karapinar andOMlesnite ldquoUlam-hyers stabilityresults for fixed point problems via 120572-120595-contractive mappingin (b)-metric spacerdquo Abstract and Applied Analysis vol 2013Article ID 825293 6 pages 2013

[4] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacerdquo Functional Analysis vol 30 pp 26ndash37 1989

[5] C Chen J Dong and C Zhu ldquoSome fixed point theorems inb-metric-like spacesrdquo Fixed Point Theory and Applications vol2015 no 122 10 pages 2015

[6] H-SDingM Imdad S Radenovic and J Vujakovic ldquoOn somefixed point results in b-metric rectangular and b-rectangularmetric spacesrdquo Arab Journal of Mathematical Sciences vol 22(2016) pp 151ndash164 2015

[7] S Radenovic M Jovanovic and Z Kadelburg ldquoCommon fixedpoint results in metric-type spacesrdquo Fixed Point Theory andApplications vol 2010 article 978121 15 pages 2010

[8] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 article 315398 2010

[9] J R Roshan V Parvaneh and Z Kadelburg ldquoCommon fixedpoint theorems for weakly isotone increasing mappings inordered b-metric spacesrdquo Journal of Nonlinear Science and itsApplications JNSA vol 7 no 4 pp 229ndash245 2014

[10] W Sintunavarat S Plubtieng and P Katchang ldquoFixed pointresult and applications on a b-metric space endowed with anarbitrary binary relationrdquo Fixed Point Theory and Applicationsvol 2013 article 296 2013

[11] W Sintunavarat ldquoGeneralized Ulam-Hyers stability well-posedness and limit shadowing of fixed point problems for 120572-120573-contraction mapping in metric spacesrdquo The Scientific WorldJournal vol 2014 Article ID 569174 7 pages 2014

[12] R J Shahkoohi and A Razani ldquoSome fixed point theoremsfor rational Geraghty contractivemappings in ordered b-metricspacesrdquo Journal of Inequalities and Applications vol 373 p 232014

[13] S L Singh and B Prasad ldquoSome coincidence theorems andstability of iterative proceduresrdquoComputersampMathematics withApplications vol 55 no 11 pp 2512ndash2520 2008

[14] M A Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[15] S Cho J Bae and E Karapınar ldquoFixed point theorems for 120572-Geraghty contraction type maps in metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 329 11 pages 2013

[16] D Dukic Z Kadelburg and S Radenovic ldquoFixed points ofGeraghty-type mappings in various generalized metric spacesrdquoAbstract and Applied Analysis vol 2011 Article ID 561245 13pages 2011

[17] M E Gordji M Ramezani Y J Cho and S Pirbavafa ldquoAgeneralization ofGeraghtyrsquos theorem in partially orderedmetricspaces and applications to ordinary differential equationsrdquoFixed Point Theory and Applications vol 2012 article 74 2012

[18] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

10 Abstract and Applied Analysis

[19] A Felhi S Sahmim and H Aydi ldquoUlam-hyers stability andwell-posedness of fixed point problems for 120572-120582-metric spacesrdquoFixed Point Theory and Applications vol 2016 no 1 20 pages2016

[20] V L Rosa and P Vetro ldquoCommon fixed points for 120572-120595-120601-contractions in generalized metric spacesrdquo Nonlinear AnalysisModelling and Control vol 19 no 1 pp 43ndash54 2014

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Abstract and Applied Analysis 5

To show 119865119911 = 119892119911 we consider120575119889(119865119911119892119911) le 120575119904119889(119865119911119865119909119899(119901))+119904119889(119892119909119899(119901)+1 119892119911)

le 120575119904119889(119865119911119865119909119899(119901)) times 120575119904119889(119892119909119899(119901)+1119892119911)le [120572 (119892119911 119892119909119899(119901)) minus 1 + 120575]119904119889(119865119911119865119909119899(119901))times 120575119904119889(119892119909119899(119901)+1119892119911)

le 120575119904120573(119889(119892119911119892119909119899(119901)))119889(119892119911119892119909119899(119901)) times 120575119904119889(119892119909119899(119901)+1 119892119911)

(35)

This in turns implies that

119889 (119865119911 119892119911) le 119904120573 (119889 (119892119911 119892119909119899(119901))) 119889 (119892119911 119892119909119899(119901))+ 119904119889 (119892119909119899(119901)+1 119892119911)

1119904 119889 (119865119911 119892119911) le 120573 (119889 (119892119911 119892119909119899(119901))) 119889 (119892119911 119892119909119899(119901))+ 119889 (119892119909119899(119901)+1 119892119911)

lt 1119904 119889 (119892119911 119892119909119899(119901)) + 119889 (119892119909119899(119901)+1 119892119911)997888rarr 0 as 119899 997888rarr infin

(36)

Thus 119865119911 = 119892119911 = 119906 is a point of coincidence of F and 119892Step 4 Next we prove that F and 119892 have a common fixedpoint Firstly we claim that if 119865119906 = 119892119906 and 119865V = 119892V then119892119906 = 119892V By hypotheses 120572(119906 V) ge 1 or 120572(V 119906) ge 1 Supposethat 120572(119906 V) ge 1 and consider that

120575119889(119892119906119892V) = 120575119889(119865119906119865V) le [120572 (119892119906 119892V) minus 1 + 120575]119889(119865119906119865V)le 120575120573(119889(119892119906119892V))119889(119892119906119892V) (37)

Then we get that

119889 (119892119906 119892V) le 120573 (119889 (119892119906 119892V)) 119889 (119892119906 119892V)119889 (119892119906 119892V) le 119889 (119892119906 119892V) (38)

which is a contradictionTherefore we conclude that 119892119906 = 119892V If we take 120572(V 119906) ge1 then we also get the same resultSecondly If F and 119892 are weakly compatible then for any119906 isin 119883 if 119906 = 119865V = 119892V (because F and 119892 have a point

of coincidence proven earlier in Step 3) then using weakcompatibility of F and 119892 we get

119865119906 = 119865 (119892V) = 119892 (119865V) = 119892119906 (39)

Thus 119906 is a coincidence point of F and 119892 then using theresult in first part 119892119906 = 119892V which leads to 119865119906 = 119892119906 = 119892V =119865V = 119906

Therefore 119906 is a common fixed point of F and 119892We can prove the uniqueness of the common fixed point

of F and 119892 by making use of condition (19) and assumption(A1) of hypothesis The proof is very simple therefore we donot go through details

If we consider 119892 as identity mapping in the abovetheorem we deduce the following corollary

Corollary 20 Let (119883 119889) be a complete b-metric space and let119865 119883 rarr 119883 be continuous 120572-admissible mapping satisfying thefollowing condition

[120572 (119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120573(119889(119909119910))119889(119909119910) (40)

for all 119909 119910 isin 119883 and 120573 isin 119861119904 where 1 lt 120575If 120572 is transitive mapping and there exists 1199090 isin 119883 such that120572(1199090 1198651199090) ge 1 then F has a fixed point

Taking 120572(119909 119910) = 1 and 120575 = 1 in Corollary 20 we get thefollowing variant of Geraghty theorem

Corollary 21 Let (119883 119889) be a complete b-metric space and let119865 119883 rarr 119883 be self-mapping satisfying the following condition

119889 (119865119909 119865119910) le 120573 (119889 (119909 119910)) 119889 (119909 119910) (41)

for all 119909 119910 isin 119883 and 120573 isin 119861119904 Then F has a unique fixed point119911 isin 119883 and for each119909 isin 119883 the Picard sequence 119865119899119909 convergesto 119911 when 119899 rarr infin

The following lemma derived from [7] is very useful toprove our next theorem

Lemma 22 (see [7]) Let 119910119899 be a sequence in a metric typespace or b-metric space (X d) such that

119889 (119910119899+1 119910119899) le 120582119889 (119910119899 119910119899minus1) (42)

for some 120582 0 lt 120582 lt 1119904 and each 119899 = 1 2 Then 119910119899 is aCauchy sequence in X

Theorem 23 Let (119883 119889) be a complete b-metric space and be120572-regular with respect to 119892 And let 119865 119892 119883 rarr 119883 be two self-mappings where F is 119892-120572-admissible and 120572 is transitive andfor 1199090 isin 119883 one has 120572(1198921199090 1198651199090) ge 1 If 119865(119909) sube 119892(119909) and oneof these two subsets of 119883 is complete suppose that there exists120582 isin [0 1119904) such that

[120572 (119892119909 119892119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119865119892119909119910) (43)

for all 119909 119910 isin 119883 and 1 lt 120575Consider

119872(119865 119892 119909 119910) = max119889 (119892119909 119892119910) 119889 (119892119909 119865119909) 119889 (119892119910 119865119910) 119889 (119892119909 119865119910) + 119889 (119892119910 119865119909)2119904

(44)

Then F and 119892 have a coincidence point

Moreover if F and 119892 are weakly compatible and hypoth-esis has one more additional assumption

(A1) either 120572(119906 V) ge 1 or 120572(V 119906) ge 1 whenever 119865119906 = 119892119906and 119865V = 119892V

then F and 119892 have a unique common fixed point

6 Abstract and Applied Analysis

Proof Let 1199090 isin 119883 be arbitrary and using condition119865(119883) sube 119892(119883) we can easily construct a Jungck sequence 119910119899satisfying

119910119899 = 119865119909119899 = 119892119909119899+1 forall119899 isin N (45)

Now if119910119899 = 119910119899+1 for any 119899 isin N then119892119909119899+1 = 119910119899 = 119910119899+1 =119865119909119899+1 and F and 119892 have a point of coincidenceWithout loss of generality we assume that 119910119899 = 119910119899+1 for

each 119899 isin NHence F is 119892-120572-admissible and 120572(1198921199090 1198921199091) = 120572(11989211990901198651199090) ge 1 Similarly 120572(1198921199091 1198921199092) = 120572(1198651199090 1198651199091) ge 1 By

induction we can easily deduce

120572 (119892119909119899minus1 119892119909119899) ge 1 forall119899 isin N (46)

Step 1 We shall show that 119910119899 is a b-Cauchy sequence Nowtaking (43) and (46) we obtain for each 119899 isin N

120575119889(119910119899+1 119910119899) = 120575119889(119865119909119899+1 119865119909119899)le [120572 (119892119909119899+1 119892119909119899) minus 1 + 120575]119889(119865119909119899+1 119865119909119899)le 120575120582119872(119865119892119909119899+1119909119899)

(47)

This implies that

119889 (119910119899+1 119910119899) le 120582119872(119865 119892 119909119899+1 119909119899) (48)

or

119889 (119910119899+1 119910119899) le 120582max119889 (119892119909119899+1 119892119909119899) 119889 (119892119909119899+1 119865119909119899+1) 119889 (119892119909119899 119865119909119899) 119889 (119892119909119899+1 119865119909119899) + 119889 (119892119909119899 119865119909119899+1)2119904

le 120582max119889 (119910119899 119910119899minus1) 119889 (119910119899 119910119899+1) 119889 (119910119899minus1 119910119899) 119889 (119910119899 119910119899) + 119889 (119910119899minus1 119910119899+1)2119904

le 120582max119889 (119910119899 119910119899minus1) 119889 (119910119899minus1 119910119899) + 119889 (119910119899 119910119899+1)2

(49)

Now if max 119889(119910119899 119910119899minus1) (119889(119910119899minus1 119910119899) + 119889(119910119899 119910119899+1))2 =(119889(119910119899minus1 119910119899)+119889(119910119899 119910119899+1))2 then 119889(119910119899 119910119899minus1) lt (119889(119910119899minus1 119910119899)+119889(119910119899 119910119899+1))2 lt 119889(119910119899+1 119910119899)Then (49) implies that

119889 (119910119899+1 119910119899) le 120582119889 (119910119899+1 119910119899) (50)

which is impossible as 120582 lt 1Therefore we deduce that max 119889(119910119899 119910119899minus1) (119889(119910119899minus1 119910119899)+119889(119910119899 119910119899+1))2 = 119889(119910119899 119910119899minus1)This in turn implies that

119889 (119910119899+1 119910119899) le 120582119889 (119910119899 119910119899minus1) (51)

Using Lemma 22 we obtain that 119910119899 is a b-Cauchysequence and by hypothesis we can suppose that 119892(119883) is tobe complete subspace of 119883 (the proof when 119865(119883) is similar)Then b-completeness of 119892(119883) implies that 119910119899 = 119865119909119899 =119892119909119899+1 b-converges to a point 119906 isin 119892(119883) where 119906 = 119892119911 forsome 119911 isin 119883 or in other words

lim119899rarrinfin

119865119909119899 = lim119899rarrinfin

119892119909119899 = 119892119911 for some 119911 isin 119883 (52)

Step 2 Next we will prove that 119865119911 = 119892119911 Since119883 is 120572-regularwith respect to 119892 and using (52) we have

120572 (119892119909119899(119901) 119892119911) ge 1 forall119901 isin N (53)

To show 119865119911 = 119892119911 we consider120575119889(119865119909119899(119901) 119865119911)le [120572 (119892119909119899(119901) 119892119911) minus 1 + 120575]119889(119865119909119899(119901) 119865119911) le 120575120582119872(119865119892119909119899(119901) 119911)

(54)

This implies that

119889 (119865119909119899(119901) 119865119911) le 120582119872(119865 119892 119909119899(119901) 119911) (55)

or

119889 (119865119909119899(119901) 119865119911) le 120582max119889 (119892119909119899(119901) 119892119911) 119889 (119892119909119899(119901) 119865119909119899(119901)) 119889 (119892119911 119865119911) 119889 (119892119909119899(119901) 119865119911) + 119889 (119892119911 119865119909119899(119901))2119904

(56)

Also119865119909119899(119901) rarr 119892119911 and 119892119909119899(119901) rarr 119892119911 as 119899 rarr infin implyingthat

119889 (119892119909119899(119901) 119865119909119899(119901)) = 119904119889 (119892119909119899(119901) 119892119911) + 119904119889 (119892119911 119865119909119899(119901))997888rarr 0 as 119899 997888rarr infin (57)

Then we have only two cases

Case 1

119889 (119865119909119899(119901) 119865119911) le 120582119889 (119892119911 119865119911)le 120582119904 (119889 (119892119911 119865119909119899(119901)) + 119889 (119865119909119899(119901) 119865119911))

(1 minus 120582119904) 119889 (119865119909119899(119901) 119865119911) le 120582119904119889 (119892119911 119865119909119899(119901)) 997888rarr 0as 119899 997888rarr infin

(58)

Abstract and Applied Analysis 7

Since 1 minus 120582119904 gt 0 it follows that 119865119909119899(119901) rarr 119865119911Case 2

119889 (119865119909119899(119901) 119865119911) le 120582119889 (119892119909119899(119901) 119865119911)2119904le 120582119904(119889 (119892119909119899(119901) 119865119909119899(119901)) + 119889 (119865119909119899(119901) 119865119911)2119904 )

(1 minus 1205822)119889 (119865119909119899(119901) 119865119911) le 1205822119889 (119892119909119899(119901) 119865119909119899(119901)) 997888rarr 0as 119899 997888rarr infin

(59)

Since 1 minus 1205822 gt 0 it follows that 119865119909119899(119901) rarr 119865119911Uniqueness of the limit of a sequence implies that 119865119911 =119892119911Using conditions (A1) and weak compatibility of F and 119892

of hypothesis we can easily prove that F and 119892 have a uniquecommon fixed point

From the above theorem one can deduce the followingcorollary easily

Corollary 24 Let (119883 119889) be a complete b-metric space and 119865 119883 rarr 119883 is continuous and 120572-admissible Suppose that thereexists 120582 isin [0 1119904) such that

[120572 (119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119909119910) (60)

for all 119909 119910 isin 119883 and 1 lt 120575Consider

119872(119909 119910) = max119889 (119909 119910) 119889 (119909 119865119909) 119889 (119910 119865119910) 119889 (119909 119865119910) + 119889 (119910 119865119909)

2119904 (61)

If 120572 is a transitive mapping and there exists 1199090 isin 119883 suchthat 120572(1199090 1198651199090) ge 1 then 119865 has a fixed point

Taking 120572(119909 119910) = 1 and 120575 = 1 we get the following variantof Corollary 312 of [7]

Corollary 25 Let (119883 119889) be a complete b-metric space and let119865 119883 rarr 119883 be self-mapping Suppose that there exists 120582 isin[0 1119904) such that for all 119909 119910 isin 119883119889 (119865119909 119865119910) le 120582119872(119909 119910) (62)

where

119872(119909 119910) = max119889 (119909 119910) 119889 (119909 119865119909) 119889 (119910 119865119910) 119889 (119909 119865119910) + 119889 (119910 119865119909)

2119904 (63)

Then F has a unique fixed point 119911 isin 119883

In what follows we furnish illustrative examples whereinone demonstrates Theorems 19 and 23 on the existence anduniqueness of a common fixed point

Example 26 Let119883 = 0 1 3 be a b-metric space withmetricd given by 119889(119909 119910) = (119909 minus 119910)2 with 119904 = 2 Consider themappings 119865 119892 119883 rarr 119883 defined by 119865(0) = 1 119865(1) = 1and 119865(3) = 1 and 119892 by 119892(0) = 1 119892(1) = 1 and 119892(3) = 3 Letus take 120573 isin 119861119904 as 120573(119905) = 12 for 119905 gt 0 and 120573(0) isin [0 12)

And 120572 119883 times 119883 rarr [0infin) by

120572 (119909 119910) = 1 when 119909 119910 ge 00 otherwise

(64)

then one can examine easily that

119889 (1198650 1198651) = 119889 (1 1) = 0120573 (119889 (1198920 1198921)) 119889 (1198920 1198921) = 120573 (119889 (1 1)) 119889 (1 1) = 0

119889 (1198650 1198653) = 119889 (1 1) = 0120573 (119889 (1198920 1198923)) 119889 (1198920 1198923) = 120573 (119889 (1 3)) 119889 (1 3) = 2

119889 (1198651 1198653) = 119889 (1 1) = 0120573 (119889 (1198921 1198923)) 119889 (1198921 1198923) = 120573 (119889 (1 3)) 119889 (1 3) = 2

(65)

Thus in all cases we get

[120572 (119892119909 119892119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120573(119889(119892119909119892119910))119889(119892119909119892119910) (66)

for all 119909 119910 isin 119883 where 1 lt 120575Also we can check that F and 119892 satisfy all the other

assumptions of Theorem 19 and thus the pair F and 119892 has aunique common fixed point 119909 = 1Example 27 Let119883 = 0 1 3 be a b-metric space with metricd given by 119889(119909 119910) = (119909 minus 119910)2 with 119904 = 2 Consider themappings 119865 119892 119883 rarr 119883 defined by 119865(0) = 0 119865(1) = 1and 119865(3) = 0 and 119892 by 119892(0) = 3 119892(1) = 1 and 119892(3) = 0 Letus take 120582 isin [0 12)

And 120572 119883 times 119883 rarr [0infin) by

120572 (119909 119910) = 1 when 119909 119910 ge 00 otherwise

(67)

Now we verify Theorem 23 by considering the followingcases

8 Abstract and Applied Analysis

Case 1 (take points 0 and 1)

119889 (1198650 1198651) = 119889 (0 1) = 1119889 (1198920 1198921) = 119889 (3 1) = 4119889 (1198920 1198650) = 119889 (3 0) = 9119889 (1198921 1198651) = 119889 (1 1) = 0119889 (1198920 1198651) + 119889 (1198921 1198650)

2 sdot 2 = 119889 (3 1) + 119889 (1 0)4 = 4 + 14= 54

119872 (119865 119892 0 1) = max119889 (1198920 1198921) 119889 (1198920 1198650) 119889 (1198921 1198651) 119889 (1198920 1198651) + 119889 (1198921 1198650)4 = 4 9 0 54= 9

(68)

Case 2 (take points 1 and 3)

119889 (1198651 1198653) = 119889 (1 1) = 0119872 (119865 119892 1 3) = max119889 (1198921 1198923) 119889 (1198921 1198651) 119889 (1198923 1198653) 119889 (1198921 1198653) + 119889 (1198923 1198651)4 = 1 0 0 24= 1

(69)

Case 3 (take points 0 and 3)

119889 (1198650 1198653) = 119889 (0 0) = 0119872 (119865 119892 0 3) = max119889 (1198920 1198923) 119889 (1198920 1198650) 119889 (1198923 1198653) 119889 (1198920 1198653) + 119889 (1198923 1198650)4 = 9 9 0 94= 9

(70)

Thus in all cases F and 119892 satisfy all the assumptions ofTheorem 23 and thus the pair F and 119892 has a unique commonfixed point 119909 = 13 An Application to an Integral Equation

This section deals with the applications of results proven inthe previous section Here we will investigate the solution ofintegral equation through our results

Consider the following integral equation

119909 (119905) = int10119870 (119905 119903 119909 (119903)) 119889119903 (71)

where119870 [0 1] times [0 1] timesR rarr R

Let 119883 = 119862[0 1] be the set of continuous real functionsdefined on [0 1] Define the b-metric by 119889(119909(119905) 119910(119905)) =max119905isin[01](|119909(119905)| + |119910(119905)|)119901 for all 119909 119910 isin 119883

Consider 119901 gt 1 Then (119883 119889) is a complete b-metric spacewith the constant 119904 = 2119901minus1119865(119909(119905)) = int1

0119870(119905 119903 119909(119903))119889119903 for all 119909 isin 119883 and for all 119905 isin[0 1] Then the existence of a solution to (71) is equivalent to

the existence of a fixed point of F Nowwe prove the followingresult

Theorem 28 Let us suppose that the following hypotheseshold

(i) 119870 [0 1] times [0 1] timesR rarr R is continuous

(ii) For all 119905 119903 isin [0 1] there exists a continuous operator120585 [0 1] times [0 1] rarr R such that

|119870 (119905 119903 119909 (119903))| + 1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816lt 1205821119901120585 (119905 119903) (|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)

sup119905isin[01]

int10120585 (119905 119903) 119889119903 le 1

(72)

where 0 lt 120582 lt 1119904(iii) There exists a function 120583 R2 rarr R such that for all119905 isin 119868 and for all 119886 119887 isin R with 120583(119886 119887) ge 0 we have

120583(1199091 (119905) int10119870 (119905 119903 119909 (119903)) 119889119903) ge 0

for 119909 isin 119862 (119868) forall119905 isin 119868120583 (119909 (119905) 119910 (119905)) ge 0

120583 (int10119870 (119905 119903 119909 (119903)) int1

0119870(119905 119903 119910 (119903)) 119889119903) ge 0

forall119905 isin 119868 forall119909 119910 isin 119862 (119868)

(73)

Then the integral equation (71) has a unique solution 119909 isin119883Proof We define 120572 119862(119868) times 119862(119868) rarr [0infin) by

120572 (119909 119910) = 1 when 120583 (119909 (119905) 119910 (119905)) ge 00 otherwise

(74)

Then for all 119909 119910 isin 119862(119868) we have 120572(119909 119910) = 1 and120572(119910 119911) = 1 implying that 120572(119909 119911) = 1 for all 119909 119910 119911 isin 119862(119868)which proves that 120572 is a transitive mapping

If 120572(119909 119910) = 1 for all 119909 119910 isin 119862(119868) then 120583(119909(119905) 119910(119905)) ge 0From (iii) we have 120583(119865119909(119905) 119865119910(119905)) ge 0 and so 120572(119865119909 119865119910) = 1Thus F is 120572-admissible

From (iii) there exists 1199091 isin 119862(119868) such that 120572(1199091 1198651199091) = 1

Abstract and Applied Analysis 9

Also we observe from (72) that

(|119865119909 (119905)| + 1003816100381610038161003816119865119910 (119905)1003816100381610038161003816)119901= (100381610038161003816100381610038161003816100381610038161003816int

1

0119870 (119905 119903 119909 (119903)) 119889119903100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816int1

0119870(119905 119903 119910 (119903)) 119889119903100381610038161003816100381610038161003816100381610038161003816)

119901

le (int10|119870 (119905 119903 119909 (119903))| 119889119903 + int1

0

1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816 119889119903)119901

le (int10(|119870 (119905 119903 119909 (119903))| + 1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816) 119889119903)

119901

le (int10(1205821119901120585 (119905 119903) (|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)) 119889119903)

119901

= (int10(1205821119901120585 (119905 119903) ((|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)119901)1119901)119889119903)

119901

le (int101205821119901120585 (119905 119903) 1198891119901 (119909 (119903) 119910 (119903)) 119889119903)119901

= 120582119889 (119909 (119905) 119910 (119905)) (int10120585 (119905 119903) 119889119903)119901

le 120582119889 (119909 (119905) 119910 (119905)) le 120582119872(119909 (119905) 119910 (119905))

(75)

which in turn implies that [120572(119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119909119910)Now all the conditions of Corollary 24 hold and F has a

unique fixed point 119909 isin 119883 which means that 119909 is the uniquesolution for the integral equation (71)

4 Conclusion

To conclude we can assert that our results are novelinteresting and generalized while considering the alpha-admissible Geraghty type mappings These results extendimprove unify and generalize many theorems based onalpha-admissible mappings in 119887-metric spaces Examples arepresented in a simplest form and illustrate the theoremsApplication to integral equation adds value to our research

Conflicts of Interest

All the authors of this article declare that they have noconflicts of interest regarding the publication of this article

Authorsrsquo Contributions

All authors have equal contribution in writing this paper Allauthors read and approved the final paper

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] R Allahyari R Arab and A S Haghighi ldquoFixed pointsof admissible almost contractive type mappings on b-metric

spaces with an application to quadratic integral equationsrdquoJournal of Inequalities and Applications vol 2015 no 32 18pages 2015

[3] M-F Bota E Karapinar andOMlesnite ldquoUlam-hyers stabilityresults for fixed point problems via 120572-120595-contractive mappingin (b)-metric spacerdquo Abstract and Applied Analysis vol 2013Article ID 825293 6 pages 2013

[4] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacerdquo Functional Analysis vol 30 pp 26ndash37 1989

[5] C Chen J Dong and C Zhu ldquoSome fixed point theorems inb-metric-like spacesrdquo Fixed Point Theory and Applications vol2015 no 122 10 pages 2015

[6] H-SDingM Imdad S Radenovic and J Vujakovic ldquoOn somefixed point results in b-metric rectangular and b-rectangularmetric spacesrdquo Arab Journal of Mathematical Sciences vol 22(2016) pp 151ndash164 2015

[7] S Radenovic M Jovanovic and Z Kadelburg ldquoCommon fixedpoint results in metric-type spacesrdquo Fixed Point Theory andApplications vol 2010 article 978121 15 pages 2010

[8] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 article 315398 2010

[9] J R Roshan V Parvaneh and Z Kadelburg ldquoCommon fixedpoint theorems for weakly isotone increasing mappings inordered b-metric spacesrdquo Journal of Nonlinear Science and itsApplications JNSA vol 7 no 4 pp 229ndash245 2014

[10] W Sintunavarat S Plubtieng and P Katchang ldquoFixed pointresult and applications on a b-metric space endowed with anarbitrary binary relationrdquo Fixed Point Theory and Applicationsvol 2013 article 296 2013

[11] W Sintunavarat ldquoGeneralized Ulam-Hyers stability well-posedness and limit shadowing of fixed point problems for 120572-120573-contraction mapping in metric spacesrdquo The Scientific WorldJournal vol 2014 Article ID 569174 7 pages 2014

[12] R J Shahkoohi and A Razani ldquoSome fixed point theoremsfor rational Geraghty contractivemappings in ordered b-metricspacesrdquo Journal of Inequalities and Applications vol 373 p 232014

[13] S L Singh and B Prasad ldquoSome coincidence theorems andstability of iterative proceduresrdquoComputersampMathematics withApplications vol 55 no 11 pp 2512ndash2520 2008

[14] M A Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[15] S Cho J Bae and E Karapınar ldquoFixed point theorems for 120572-Geraghty contraction type maps in metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 329 11 pages 2013

[16] D Dukic Z Kadelburg and S Radenovic ldquoFixed points ofGeraghty-type mappings in various generalized metric spacesrdquoAbstract and Applied Analysis vol 2011 Article ID 561245 13pages 2011

[17] M E Gordji M Ramezani Y J Cho and S Pirbavafa ldquoAgeneralization ofGeraghtyrsquos theorem in partially orderedmetricspaces and applications to ordinary differential equationsrdquoFixed Point Theory and Applications vol 2012 article 74 2012

[18] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

10 Abstract and Applied Analysis

[19] A Felhi S Sahmim and H Aydi ldquoUlam-hyers stability andwell-posedness of fixed point problems for 120572-120582-metric spacesrdquoFixed Point Theory and Applications vol 2016 no 1 20 pages2016

[20] V L Rosa and P Vetro ldquoCommon fixed points for 120572-120595-120601-contractions in generalized metric spacesrdquo Nonlinear AnalysisModelling and Control vol 19 no 1 pp 43ndash54 2014

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Proof Let 1199090 isin 119883 be arbitrary and using condition119865(119883) sube 119892(119883) we can easily construct a Jungck sequence 119910119899satisfying

119910119899 = 119865119909119899 = 119892119909119899+1 forall119899 isin N (45)

Now if119910119899 = 119910119899+1 for any 119899 isin N then119892119909119899+1 = 119910119899 = 119910119899+1 =119865119909119899+1 and F and 119892 have a point of coincidenceWithout loss of generality we assume that 119910119899 = 119910119899+1 for

each 119899 isin NHence F is 119892-120572-admissible and 120572(1198921199090 1198921199091) = 120572(11989211990901198651199090) ge 1 Similarly 120572(1198921199091 1198921199092) = 120572(1198651199090 1198651199091) ge 1 By

induction we can easily deduce

120572 (119892119909119899minus1 119892119909119899) ge 1 forall119899 isin N (46)

Step 1 We shall show that 119910119899 is a b-Cauchy sequence Nowtaking (43) and (46) we obtain for each 119899 isin N

120575119889(119910119899+1 119910119899) = 120575119889(119865119909119899+1 119865119909119899)le [120572 (119892119909119899+1 119892119909119899) minus 1 + 120575]119889(119865119909119899+1 119865119909119899)le 120575120582119872(119865119892119909119899+1119909119899)

(47)

This implies that

119889 (119910119899+1 119910119899) le 120582119872(119865 119892 119909119899+1 119909119899) (48)

or

119889 (119910119899+1 119910119899) le 120582max119889 (119892119909119899+1 119892119909119899) 119889 (119892119909119899+1 119865119909119899+1) 119889 (119892119909119899 119865119909119899) 119889 (119892119909119899+1 119865119909119899) + 119889 (119892119909119899 119865119909119899+1)2119904

le 120582max119889 (119910119899 119910119899minus1) 119889 (119910119899 119910119899+1) 119889 (119910119899minus1 119910119899) 119889 (119910119899 119910119899) + 119889 (119910119899minus1 119910119899+1)2119904

le 120582max119889 (119910119899 119910119899minus1) 119889 (119910119899minus1 119910119899) + 119889 (119910119899 119910119899+1)2

(49)

Now if max 119889(119910119899 119910119899minus1) (119889(119910119899minus1 119910119899) + 119889(119910119899 119910119899+1))2 =(119889(119910119899minus1 119910119899)+119889(119910119899 119910119899+1))2 then 119889(119910119899 119910119899minus1) lt (119889(119910119899minus1 119910119899)+119889(119910119899 119910119899+1))2 lt 119889(119910119899+1 119910119899)Then (49) implies that

119889 (119910119899+1 119910119899) le 120582119889 (119910119899+1 119910119899) (50)

which is impossible as 120582 lt 1Therefore we deduce that max 119889(119910119899 119910119899minus1) (119889(119910119899minus1 119910119899)+119889(119910119899 119910119899+1))2 = 119889(119910119899 119910119899minus1)This in turn implies that

119889 (119910119899+1 119910119899) le 120582119889 (119910119899 119910119899minus1) (51)

Using Lemma 22 we obtain that 119910119899 is a b-Cauchysequence and by hypothesis we can suppose that 119892(119883) is tobe complete subspace of 119883 (the proof when 119865(119883) is similar)Then b-completeness of 119892(119883) implies that 119910119899 = 119865119909119899 =119892119909119899+1 b-converges to a point 119906 isin 119892(119883) where 119906 = 119892119911 forsome 119911 isin 119883 or in other words

lim119899rarrinfin

119865119909119899 = lim119899rarrinfin

119892119909119899 = 119892119911 for some 119911 isin 119883 (52)

Step 2 Next we will prove that 119865119911 = 119892119911 Since119883 is 120572-regularwith respect to 119892 and using (52) we have

120572 (119892119909119899(119901) 119892119911) ge 1 forall119901 isin N (53)

To show 119865119911 = 119892119911 we consider120575119889(119865119909119899(119901) 119865119911)le [120572 (119892119909119899(119901) 119892119911) minus 1 + 120575]119889(119865119909119899(119901) 119865119911) le 120575120582119872(119865119892119909119899(119901) 119911)

(54)

This implies that

119889 (119865119909119899(119901) 119865119911) le 120582119872(119865 119892 119909119899(119901) 119911) (55)

or

119889 (119865119909119899(119901) 119865119911) le 120582max119889 (119892119909119899(119901) 119892119911) 119889 (119892119909119899(119901) 119865119909119899(119901)) 119889 (119892119911 119865119911) 119889 (119892119909119899(119901) 119865119911) + 119889 (119892119911 119865119909119899(119901))2119904

(56)

Also119865119909119899(119901) rarr 119892119911 and 119892119909119899(119901) rarr 119892119911 as 119899 rarr infin implyingthat

119889 (119892119909119899(119901) 119865119909119899(119901)) = 119904119889 (119892119909119899(119901) 119892119911) + 119904119889 (119892119911 119865119909119899(119901))997888rarr 0 as 119899 997888rarr infin (57)

Then we have only two cases

Case 1

119889 (119865119909119899(119901) 119865119911) le 120582119889 (119892119911 119865119911)le 120582119904 (119889 (119892119911 119865119909119899(119901)) + 119889 (119865119909119899(119901) 119865119911))

(1 minus 120582119904) 119889 (119865119909119899(119901) 119865119911) le 120582119904119889 (119892119911 119865119909119899(119901)) 997888rarr 0as 119899 997888rarr infin

(58)

Abstract and Applied Analysis 7

Since 1 minus 120582119904 gt 0 it follows that 119865119909119899(119901) rarr 119865119911Case 2

119889 (119865119909119899(119901) 119865119911) le 120582119889 (119892119909119899(119901) 119865119911)2119904le 120582119904(119889 (119892119909119899(119901) 119865119909119899(119901)) + 119889 (119865119909119899(119901) 119865119911)2119904 )

(1 minus 1205822)119889 (119865119909119899(119901) 119865119911) le 1205822119889 (119892119909119899(119901) 119865119909119899(119901)) 997888rarr 0as 119899 997888rarr infin

(59)

Since 1 minus 1205822 gt 0 it follows that 119865119909119899(119901) rarr 119865119911Uniqueness of the limit of a sequence implies that 119865119911 =119892119911Using conditions (A1) and weak compatibility of F and 119892

of hypothesis we can easily prove that F and 119892 have a uniquecommon fixed point

From the above theorem one can deduce the followingcorollary easily

Corollary 24 Let (119883 119889) be a complete b-metric space and 119865 119883 rarr 119883 is continuous and 120572-admissible Suppose that thereexists 120582 isin [0 1119904) such that

[120572 (119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119909119910) (60)

for all 119909 119910 isin 119883 and 1 lt 120575Consider

119872(119909 119910) = max119889 (119909 119910) 119889 (119909 119865119909) 119889 (119910 119865119910) 119889 (119909 119865119910) + 119889 (119910 119865119909)

2119904 (61)

If 120572 is a transitive mapping and there exists 1199090 isin 119883 suchthat 120572(1199090 1198651199090) ge 1 then 119865 has a fixed point

Taking 120572(119909 119910) = 1 and 120575 = 1 we get the following variantof Corollary 312 of [7]

Corollary 25 Let (119883 119889) be a complete b-metric space and let119865 119883 rarr 119883 be self-mapping Suppose that there exists 120582 isin[0 1119904) such that for all 119909 119910 isin 119883119889 (119865119909 119865119910) le 120582119872(119909 119910) (62)

where

119872(119909 119910) = max119889 (119909 119910) 119889 (119909 119865119909) 119889 (119910 119865119910) 119889 (119909 119865119910) + 119889 (119910 119865119909)

2119904 (63)

Then F has a unique fixed point 119911 isin 119883

In what follows we furnish illustrative examples whereinone demonstrates Theorems 19 and 23 on the existence anduniqueness of a common fixed point

Example 26 Let119883 = 0 1 3 be a b-metric space withmetricd given by 119889(119909 119910) = (119909 minus 119910)2 with 119904 = 2 Consider themappings 119865 119892 119883 rarr 119883 defined by 119865(0) = 1 119865(1) = 1and 119865(3) = 1 and 119892 by 119892(0) = 1 119892(1) = 1 and 119892(3) = 3 Letus take 120573 isin 119861119904 as 120573(119905) = 12 for 119905 gt 0 and 120573(0) isin [0 12)

And 120572 119883 times 119883 rarr [0infin) by

120572 (119909 119910) = 1 when 119909 119910 ge 00 otherwise

(64)

then one can examine easily that

119889 (1198650 1198651) = 119889 (1 1) = 0120573 (119889 (1198920 1198921)) 119889 (1198920 1198921) = 120573 (119889 (1 1)) 119889 (1 1) = 0

119889 (1198650 1198653) = 119889 (1 1) = 0120573 (119889 (1198920 1198923)) 119889 (1198920 1198923) = 120573 (119889 (1 3)) 119889 (1 3) = 2

119889 (1198651 1198653) = 119889 (1 1) = 0120573 (119889 (1198921 1198923)) 119889 (1198921 1198923) = 120573 (119889 (1 3)) 119889 (1 3) = 2

(65)

Thus in all cases we get

[120572 (119892119909 119892119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120573(119889(119892119909119892119910))119889(119892119909119892119910) (66)

for all 119909 119910 isin 119883 where 1 lt 120575Also we can check that F and 119892 satisfy all the other

assumptions of Theorem 19 and thus the pair F and 119892 has aunique common fixed point 119909 = 1Example 27 Let119883 = 0 1 3 be a b-metric space with metricd given by 119889(119909 119910) = (119909 minus 119910)2 with 119904 = 2 Consider themappings 119865 119892 119883 rarr 119883 defined by 119865(0) = 0 119865(1) = 1and 119865(3) = 0 and 119892 by 119892(0) = 3 119892(1) = 1 and 119892(3) = 0 Letus take 120582 isin [0 12)

And 120572 119883 times 119883 rarr [0infin) by

120572 (119909 119910) = 1 when 119909 119910 ge 00 otherwise

(67)

Now we verify Theorem 23 by considering the followingcases

8 Abstract and Applied Analysis

Case 1 (take points 0 and 1)

119889 (1198650 1198651) = 119889 (0 1) = 1119889 (1198920 1198921) = 119889 (3 1) = 4119889 (1198920 1198650) = 119889 (3 0) = 9119889 (1198921 1198651) = 119889 (1 1) = 0119889 (1198920 1198651) + 119889 (1198921 1198650)

2 sdot 2 = 119889 (3 1) + 119889 (1 0)4 = 4 + 14= 54

119872 (119865 119892 0 1) = max119889 (1198920 1198921) 119889 (1198920 1198650) 119889 (1198921 1198651) 119889 (1198920 1198651) + 119889 (1198921 1198650)4 = 4 9 0 54= 9

(68)

Case 2 (take points 1 and 3)

119889 (1198651 1198653) = 119889 (1 1) = 0119872 (119865 119892 1 3) = max119889 (1198921 1198923) 119889 (1198921 1198651) 119889 (1198923 1198653) 119889 (1198921 1198653) + 119889 (1198923 1198651)4 = 1 0 0 24= 1

(69)

Case 3 (take points 0 and 3)

119889 (1198650 1198653) = 119889 (0 0) = 0119872 (119865 119892 0 3) = max119889 (1198920 1198923) 119889 (1198920 1198650) 119889 (1198923 1198653) 119889 (1198920 1198653) + 119889 (1198923 1198650)4 = 9 9 0 94= 9

(70)

Thus in all cases F and 119892 satisfy all the assumptions ofTheorem 23 and thus the pair F and 119892 has a unique commonfixed point 119909 = 13 An Application to an Integral Equation

This section deals with the applications of results proven inthe previous section Here we will investigate the solution ofintegral equation through our results

Consider the following integral equation

119909 (119905) = int10119870 (119905 119903 119909 (119903)) 119889119903 (71)

where119870 [0 1] times [0 1] timesR rarr R

Let 119883 = 119862[0 1] be the set of continuous real functionsdefined on [0 1] Define the b-metric by 119889(119909(119905) 119910(119905)) =max119905isin[01](|119909(119905)| + |119910(119905)|)119901 for all 119909 119910 isin 119883

Consider 119901 gt 1 Then (119883 119889) is a complete b-metric spacewith the constant 119904 = 2119901minus1119865(119909(119905)) = int1

0119870(119905 119903 119909(119903))119889119903 for all 119909 isin 119883 and for all 119905 isin[0 1] Then the existence of a solution to (71) is equivalent to

the existence of a fixed point of F Nowwe prove the followingresult

Theorem 28 Let us suppose that the following hypotheseshold

(i) 119870 [0 1] times [0 1] timesR rarr R is continuous

(ii) For all 119905 119903 isin [0 1] there exists a continuous operator120585 [0 1] times [0 1] rarr R such that

|119870 (119905 119903 119909 (119903))| + 1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816lt 1205821119901120585 (119905 119903) (|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)

sup119905isin[01]

int10120585 (119905 119903) 119889119903 le 1

(72)

where 0 lt 120582 lt 1119904(iii) There exists a function 120583 R2 rarr R such that for all119905 isin 119868 and for all 119886 119887 isin R with 120583(119886 119887) ge 0 we have

120583(1199091 (119905) int10119870 (119905 119903 119909 (119903)) 119889119903) ge 0

for 119909 isin 119862 (119868) forall119905 isin 119868120583 (119909 (119905) 119910 (119905)) ge 0

120583 (int10119870 (119905 119903 119909 (119903)) int1

0119870(119905 119903 119910 (119903)) 119889119903) ge 0

forall119905 isin 119868 forall119909 119910 isin 119862 (119868)

(73)

Then the integral equation (71) has a unique solution 119909 isin119883Proof We define 120572 119862(119868) times 119862(119868) rarr [0infin) by

120572 (119909 119910) = 1 when 120583 (119909 (119905) 119910 (119905)) ge 00 otherwise

(74)

Then for all 119909 119910 isin 119862(119868) we have 120572(119909 119910) = 1 and120572(119910 119911) = 1 implying that 120572(119909 119911) = 1 for all 119909 119910 119911 isin 119862(119868)which proves that 120572 is a transitive mapping

If 120572(119909 119910) = 1 for all 119909 119910 isin 119862(119868) then 120583(119909(119905) 119910(119905)) ge 0From (iii) we have 120583(119865119909(119905) 119865119910(119905)) ge 0 and so 120572(119865119909 119865119910) = 1Thus F is 120572-admissible

From (iii) there exists 1199091 isin 119862(119868) such that 120572(1199091 1198651199091) = 1

Abstract and Applied Analysis 9

Also we observe from (72) that

(|119865119909 (119905)| + 1003816100381610038161003816119865119910 (119905)1003816100381610038161003816)119901= (100381610038161003816100381610038161003816100381610038161003816int

1

0119870 (119905 119903 119909 (119903)) 119889119903100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816int1

0119870(119905 119903 119910 (119903)) 119889119903100381610038161003816100381610038161003816100381610038161003816)

119901

le (int10|119870 (119905 119903 119909 (119903))| 119889119903 + int1

0

1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816 119889119903)119901

le (int10(|119870 (119905 119903 119909 (119903))| + 1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816) 119889119903)

119901

le (int10(1205821119901120585 (119905 119903) (|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)) 119889119903)

119901

= (int10(1205821119901120585 (119905 119903) ((|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)119901)1119901)119889119903)

119901

le (int101205821119901120585 (119905 119903) 1198891119901 (119909 (119903) 119910 (119903)) 119889119903)119901

= 120582119889 (119909 (119905) 119910 (119905)) (int10120585 (119905 119903) 119889119903)119901

le 120582119889 (119909 (119905) 119910 (119905)) le 120582119872(119909 (119905) 119910 (119905))

(75)

which in turn implies that [120572(119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119909119910)Now all the conditions of Corollary 24 hold and F has a

unique fixed point 119909 isin 119883 which means that 119909 is the uniquesolution for the integral equation (71)

4 Conclusion

To conclude we can assert that our results are novelinteresting and generalized while considering the alpha-admissible Geraghty type mappings These results extendimprove unify and generalize many theorems based onalpha-admissible mappings in 119887-metric spaces Examples arepresented in a simplest form and illustrate the theoremsApplication to integral equation adds value to our research

Conflicts of Interest

All the authors of this article declare that they have noconflicts of interest regarding the publication of this article

Authorsrsquo Contributions

All authors have equal contribution in writing this paper Allauthors read and approved the final paper

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] R Allahyari R Arab and A S Haghighi ldquoFixed pointsof admissible almost contractive type mappings on b-metric

spaces with an application to quadratic integral equationsrdquoJournal of Inequalities and Applications vol 2015 no 32 18pages 2015

[3] M-F Bota E Karapinar andOMlesnite ldquoUlam-hyers stabilityresults for fixed point problems via 120572-120595-contractive mappingin (b)-metric spacerdquo Abstract and Applied Analysis vol 2013Article ID 825293 6 pages 2013

[4] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacerdquo Functional Analysis vol 30 pp 26ndash37 1989

[5] C Chen J Dong and C Zhu ldquoSome fixed point theorems inb-metric-like spacesrdquo Fixed Point Theory and Applications vol2015 no 122 10 pages 2015

[6] H-SDingM Imdad S Radenovic and J Vujakovic ldquoOn somefixed point results in b-metric rectangular and b-rectangularmetric spacesrdquo Arab Journal of Mathematical Sciences vol 22(2016) pp 151ndash164 2015

[7] S Radenovic M Jovanovic and Z Kadelburg ldquoCommon fixedpoint results in metric-type spacesrdquo Fixed Point Theory andApplications vol 2010 article 978121 15 pages 2010

[8] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 article 315398 2010

[9] J R Roshan V Parvaneh and Z Kadelburg ldquoCommon fixedpoint theorems for weakly isotone increasing mappings inordered b-metric spacesrdquo Journal of Nonlinear Science and itsApplications JNSA vol 7 no 4 pp 229ndash245 2014

[10] W Sintunavarat S Plubtieng and P Katchang ldquoFixed pointresult and applications on a b-metric space endowed with anarbitrary binary relationrdquo Fixed Point Theory and Applicationsvol 2013 article 296 2013

[11] W Sintunavarat ldquoGeneralized Ulam-Hyers stability well-posedness and limit shadowing of fixed point problems for 120572-120573-contraction mapping in metric spacesrdquo The Scientific WorldJournal vol 2014 Article ID 569174 7 pages 2014

[12] R J Shahkoohi and A Razani ldquoSome fixed point theoremsfor rational Geraghty contractivemappings in ordered b-metricspacesrdquo Journal of Inequalities and Applications vol 373 p 232014

[13] S L Singh and B Prasad ldquoSome coincidence theorems andstability of iterative proceduresrdquoComputersampMathematics withApplications vol 55 no 11 pp 2512ndash2520 2008

[14] M A Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[15] S Cho J Bae and E Karapınar ldquoFixed point theorems for 120572-Geraghty contraction type maps in metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 329 11 pages 2013

[16] D Dukic Z Kadelburg and S Radenovic ldquoFixed points ofGeraghty-type mappings in various generalized metric spacesrdquoAbstract and Applied Analysis vol 2011 Article ID 561245 13pages 2011

[17] M E Gordji M Ramezani Y J Cho and S Pirbavafa ldquoAgeneralization ofGeraghtyrsquos theorem in partially orderedmetricspaces and applications to ordinary differential equationsrdquoFixed Point Theory and Applications vol 2012 article 74 2012

[18] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

10 Abstract and Applied Analysis

[19] A Felhi S Sahmim and H Aydi ldquoUlam-hyers stability andwell-posedness of fixed point problems for 120572-120582-metric spacesrdquoFixed Point Theory and Applications vol 2016 no 1 20 pages2016

[20] V L Rosa and P Vetro ldquoCommon fixed points for 120572-120595-120601-contractions in generalized metric spacesrdquo Nonlinear AnalysisModelling and Control vol 19 no 1 pp 43ndash54 2014

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

Since 1 minus 120582119904 gt 0 it follows that 119865119909119899(119901) rarr 119865119911Case 2

119889 (119865119909119899(119901) 119865119911) le 120582119889 (119892119909119899(119901) 119865119911)2119904le 120582119904(119889 (119892119909119899(119901) 119865119909119899(119901)) + 119889 (119865119909119899(119901) 119865119911)2119904 )

(1 minus 1205822)119889 (119865119909119899(119901) 119865119911) le 1205822119889 (119892119909119899(119901) 119865119909119899(119901)) 997888rarr 0as 119899 997888rarr infin

(59)

Since 1 minus 1205822 gt 0 it follows that 119865119909119899(119901) rarr 119865119911Uniqueness of the limit of a sequence implies that 119865119911 =119892119911Using conditions (A1) and weak compatibility of F and 119892

of hypothesis we can easily prove that F and 119892 have a uniquecommon fixed point

From the above theorem one can deduce the followingcorollary easily

Corollary 24 Let (119883 119889) be a complete b-metric space and 119865 119883 rarr 119883 is continuous and 120572-admissible Suppose that thereexists 120582 isin [0 1119904) such that

[120572 (119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119909119910) (60)

for all 119909 119910 isin 119883 and 1 lt 120575Consider

119872(119909 119910) = max119889 (119909 119910) 119889 (119909 119865119909) 119889 (119910 119865119910) 119889 (119909 119865119910) + 119889 (119910 119865119909)

2119904 (61)

If 120572 is a transitive mapping and there exists 1199090 isin 119883 suchthat 120572(1199090 1198651199090) ge 1 then 119865 has a fixed point

Taking 120572(119909 119910) = 1 and 120575 = 1 we get the following variantof Corollary 312 of [7]

Corollary 25 Let (119883 119889) be a complete b-metric space and let119865 119883 rarr 119883 be self-mapping Suppose that there exists 120582 isin[0 1119904) such that for all 119909 119910 isin 119883119889 (119865119909 119865119910) le 120582119872(119909 119910) (62)

where

119872(119909 119910) = max119889 (119909 119910) 119889 (119909 119865119909) 119889 (119910 119865119910) 119889 (119909 119865119910) + 119889 (119910 119865119909)

2119904 (63)

Then F has a unique fixed point 119911 isin 119883

In what follows we furnish illustrative examples whereinone demonstrates Theorems 19 and 23 on the existence anduniqueness of a common fixed point

Example 26 Let119883 = 0 1 3 be a b-metric space withmetricd given by 119889(119909 119910) = (119909 minus 119910)2 with 119904 = 2 Consider themappings 119865 119892 119883 rarr 119883 defined by 119865(0) = 1 119865(1) = 1and 119865(3) = 1 and 119892 by 119892(0) = 1 119892(1) = 1 and 119892(3) = 3 Letus take 120573 isin 119861119904 as 120573(119905) = 12 for 119905 gt 0 and 120573(0) isin [0 12)

And 120572 119883 times 119883 rarr [0infin) by

120572 (119909 119910) = 1 when 119909 119910 ge 00 otherwise

(64)

then one can examine easily that

119889 (1198650 1198651) = 119889 (1 1) = 0120573 (119889 (1198920 1198921)) 119889 (1198920 1198921) = 120573 (119889 (1 1)) 119889 (1 1) = 0

119889 (1198650 1198653) = 119889 (1 1) = 0120573 (119889 (1198920 1198923)) 119889 (1198920 1198923) = 120573 (119889 (1 3)) 119889 (1 3) = 2

119889 (1198651 1198653) = 119889 (1 1) = 0120573 (119889 (1198921 1198923)) 119889 (1198921 1198923) = 120573 (119889 (1 3)) 119889 (1 3) = 2

(65)

Thus in all cases we get

[120572 (119892119909 119892119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120573(119889(119892119909119892119910))119889(119892119909119892119910) (66)

for all 119909 119910 isin 119883 where 1 lt 120575Also we can check that F and 119892 satisfy all the other

assumptions of Theorem 19 and thus the pair F and 119892 has aunique common fixed point 119909 = 1Example 27 Let119883 = 0 1 3 be a b-metric space with metricd given by 119889(119909 119910) = (119909 minus 119910)2 with 119904 = 2 Consider themappings 119865 119892 119883 rarr 119883 defined by 119865(0) = 0 119865(1) = 1and 119865(3) = 0 and 119892 by 119892(0) = 3 119892(1) = 1 and 119892(3) = 0 Letus take 120582 isin [0 12)

And 120572 119883 times 119883 rarr [0infin) by

120572 (119909 119910) = 1 when 119909 119910 ge 00 otherwise

(67)

Now we verify Theorem 23 by considering the followingcases

8 Abstract and Applied Analysis

Case 1 (take points 0 and 1)

119889 (1198650 1198651) = 119889 (0 1) = 1119889 (1198920 1198921) = 119889 (3 1) = 4119889 (1198920 1198650) = 119889 (3 0) = 9119889 (1198921 1198651) = 119889 (1 1) = 0119889 (1198920 1198651) + 119889 (1198921 1198650)

2 sdot 2 = 119889 (3 1) + 119889 (1 0)4 = 4 + 14= 54

119872 (119865 119892 0 1) = max119889 (1198920 1198921) 119889 (1198920 1198650) 119889 (1198921 1198651) 119889 (1198920 1198651) + 119889 (1198921 1198650)4 = 4 9 0 54= 9

(68)

Case 2 (take points 1 and 3)

119889 (1198651 1198653) = 119889 (1 1) = 0119872 (119865 119892 1 3) = max119889 (1198921 1198923) 119889 (1198921 1198651) 119889 (1198923 1198653) 119889 (1198921 1198653) + 119889 (1198923 1198651)4 = 1 0 0 24= 1

(69)

Case 3 (take points 0 and 3)

119889 (1198650 1198653) = 119889 (0 0) = 0119872 (119865 119892 0 3) = max119889 (1198920 1198923) 119889 (1198920 1198650) 119889 (1198923 1198653) 119889 (1198920 1198653) + 119889 (1198923 1198650)4 = 9 9 0 94= 9

(70)

Thus in all cases F and 119892 satisfy all the assumptions ofTheorem 23 and thus the pair F and 119892 has a unique commonfixed point 119909 = 13 An Application to an Integral Equation

This section deals with the applications of results proven inthe previous section Here we will investigate the solution ofintegral equation through our results

Consider the following integral equation

119909 (119905) = int10119870 (119905 119903 119909 (119903)) 119889119903 (71)

where119870 [0 1] times [0 1] timesR rarr R

Let 119883 = 119862[0 1] be the set of continuous real functionsdefined on [0 1] Define the b-metric by 119889(119909(119905) 119910(119905)) =max119905isin[01](|119909(119905)| + |119910(119905)|)119901 for all 119909 119910 isin 119883

Consider 119901 gt 1 Then (119883 119889) is a complete b-metric spacewith the constant 119904 = 2119901minus1119865(119909(119905)) = int1

0119870(119905 119903 119909(119903))119889119903 for all 119909 isin 119883 and for all 119905 isin[0 1] Then the existence of a solution to (71) is equivalent to

the existence of a fixed point of F Nowwe prove the followingresult

Theorem 28 Let us suppose that the following hypotheseshold

(i) 119870 [0 1] times [0 1] timesR rarr R is continuous

(ii) For all 119905 119903 isin [0 1] there exists a continuous operator120585 [0 1] times [0 1] rarr R such that

|119870 (119905 119903 119909 (119903))| + 1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816lt 1205821119901120585 (119905 119903) (|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)

sup119905isin[01]

int10120585 (119905 119903) 119889119903 le 1

(72)

where 0 lt 120582 lt 1119904(iii) There exists a function 120583 R2 rarr R such that for all119905 isin 119868 and for all 119886 119887 isin R with 120583(119886 119887) ge 0 we have

120583(1199091 (119905) int10119870 (119905 119903 119909 (119903)) 119889119903) ge 0

for 119909 isin 119862 (119868) forall119905 isin 119868120583 (119909 (119905) 119910 (119905)) ge 0

120583 (int10119870 (119905 119903 119909 (119903)) int1

0119870(119905 119903 119910 (119903)) 119889119903) ge 0

forall119905 isin 119868 forall119909 119910 isin 119862 (119868)

(73)

Then the integral equation (71) has a unique solution 119909 isin119883Proof We define 120572 119862(119868) times 119862(119868) rarr [0infin) by

120572 (119909 119910) = 1 when 120583 (119909 (119905) 119910 (119905)) ge 00 otherwise

(74)

Then for all 119909 119910 isin 119862(119868) we have 120572(119909 119910) = 1 and120572(119910 119911) = 1 implying that 120572(119909 119911) = 1 for all 119909 119910 119911 isin 119862(119868)which proves that 120572 is a transitive mapping

If 120572(119909 119910) = 1 for all 119909 119910 isin 119862(119868) then 120583(119909(119905) 119910(119905)) ge 0From (iii) we have 120583(119865119909(119905) 119865119910(119905)) ge 0 and so 120572(119865119909 119865119910) = 1Thus F is 120572-admissible

From (iii) there exists 1199091 isin 119862(119868) such that 120572(1199091 1198651199091) = 1

Abstract and Applied Analysis 9

Also we observe from (72) that

(|119865119909 (119905)| + 1003816100381610038161003816119865119910 (119905)1003816100381610038161003816)119901= (100381610038161003816100381610038161003816100381610038161003816int

1

0119870 (119905 119903 119909 (119903)) 119889119903100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816int1

0119870(119905 119903 119910 (119903)) 119889119903100381610038161003816100381610038161003816100381610038161003816)

119901

le (int10|119870 (119905 119903 119909 (119903))| 119889119903 + int1

0

1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816 119889119903)119901

le (int10(|119870 (119905 119903 119909 (119903))| + 1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816) 119889119903)

119901

le (int10(1205821119901120585 (119905 119903) (|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)) 119889119903)

119901

= (int10(1205821119901120585 (119905 119903) ((|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)119901)1119901)119889119903)

119901

le (int101205821119901120585 (119905 119903) 1198891119901 (119909 (119903) 119910 (119903)) 119889119903)119901

= 120582119889 (119909 (119905) 119910 (119905)) (int10120585 (119905 119903) 119889119903)119901

le 120582119889 (119909 (119905) 119910 (119905)) le 120582119872(119909 (119905) 119910 (119905))

(75)

which in turn implies that [120572(119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119909119910)Now all the conditions of Corollary 24 hold and F has a

unique fixed point 119909 isin 119883 which means that 119909 is the uniquesolution for the integral equation (71)

4 Conclusion

To conclude we can assert that our results are novelinteresting and generalized while considering the alpha-admissible Geraghty type mappings These results extendimprove unify and generalize many theorems based onalpha-admissible mappings in 119887-metric spaces Examples arepresented in a simplest form and illustrate the theoremsApplication to integral equation adds value to our research

Conflicts of Interest

All the authors of this article declare that they have noconflicts of interest regarding the publication of this article

Authorsrsquo Contributions

All authors have equal contribution in writing this paper Allauthors read and approved the final paper

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] R Allahyari R Arab and A S Haghighi ldquoFixed pointsof admissible almost contractive type mappings on b-metric

spaces with an application to quadratic integral equationsrdquoJournal of Inequalities and Applications vol 2015 no 32 18pages 2015

[3] M-F Bota E Karapinar andOMlesnite ldquoUlam-hyers stabilityresults for fixed point problems via 120572-120595-contractive mappingin (b)-metric spacerdquo Abstract and Applied Analysis vol 2013Article ID 825293 6 pages 2013

[4] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacerdquo Functional Analysis vol 30 pp 26ndash37 1989

[5] C Chen J Dong and C Zhu ldquoSome fixed point theorems inb-metric-like spacesrdquo Fixed Point Theory and Applications vol2015 no 122 10 pages 2015

[6] H-SDingM Imdad S Radenovic and J Vujakovic ldquoOn somefixed point results in b-metric rectangular and b-rectangularmetric spacesrdquo Arab Journal of Mathematical Sciences vol 22(2016) pp 151ndash164 2015

[7] S Radenovic M Jovanovic and Z Kadelburg ldquoCommon fixedpoint results in metric-type spacesrdquo Fixed Point Theory andApplications vol 2010 article 978121 15 pages 2010

[8] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 article 315398 2010

[9] J R Roshan V Parvaneh and Z Kadelburg ldquoCommon fixedpoint theorems for weakly isotone increasing mappings inordered b-metric spacesrdquo Journal of Nonlinear Science and itsApplications JNSA vol 7 no 4 pp 229ndash245 2014

[10] W Sintunavarat S Plubtieng and P Katchang ldquoFixed pointresult and applications on a b-metric space endowed with anarbitrary binary relationrdquo Fixed Point Theory and Applicationsvol 2013 article 296 2013

[11] W Sintunavarat ldquoGeneralized Ulam-Hyers stability well-posedness and limit shadowing of fixed point problems for 120572-120573-contraction mapping in metric spacesrdquo The Scientific WorldJournal vol 2014 Article ID 569174 7 pages 2014

[12] R J Shahkoohi and A Razani ldquoSome fixed point theoremsfor rational Geraghty contractivemappings in ordered b-metricspacesrdquo Journal of Inequalities and Applications vol 373 p 232014

[13] S L Singh and B Prasad ldquoSome coincidence theorems andstability of iterative proceduresrdquoComputersampMathematics withApplications vol 55 no 11 pp 2512ndash2520 2008

[14] M A Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[15] S Cho J Bae and E Karapınar ldquoFixed point theorems for 120572-Geraghty contraction type maps in metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 329 11 pages 2013

[16] D Dukic Z Kadelburg and S Radenovic ldquoFixed points ofGeraghty-type mappings in various generalized metric spacesrdquoAbstract and Applied Analysis vol 2011 Article ID 561245 13pages 2011

[17] M E Gordji M Ramezani Y J Cho and S Pirbavafa ldquoAgeneralization ofGeraghtyrsquos theorem in partially orderedmetricspaces and applications to ordinary differential equationsrdquoFixed Point Theory and Applications vol 2012 article 74 2012

[18] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

10 Abstract and Applied Analysis

[19] A Felhi S Sahmim and H Aydi ldquoUlam-hyers stability andwell-posedness of fixed point problems for 120572-120582-metric spacesrdquoFixed Point Theory and Applications vol 2016 no 1 20 pages2016

[20] V L Rosa and P Vetro ldquoCommon fixed points for 120572-120595-120601-contractions in generalized metric spacesrdquo Nonlinear AnalysisModelling and Control vol 19 no 1 pp 43ndash54 2014

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

Case 1 (take points 0 and 1)

119889 (1198650 1198651) = 119889 (0 1) = 1119889 (1198920 1198921) = 119889 (3 1) = 4119889 (1198920 1198650) = 119889 (3 0) = 9119889 (1198921 1198651) = 119889 (1 1) = 0119889 (1198920 1198651) + 119889 (1198921 1198650)

2 sdot 2 = 119889 (3 1) + 119889 (1 0)4 = 4 + 14= 54

119872 (119865 119892 0 1) = max119889 (1198920 1198921) 119889 (1198920 1198650) 119889 (1198921 1198651) 119889 (1198920 1198651) + 119889 (1198921 1198650)4 = 4 9 0 54= 9

(68)

Case 2 (take points 1 and 3)

119889 (1198651 1198653) = 119889 (1 1) = 0119872 (119865 119892 1 3) = max119889 (1198921 1198923) 119889 (1198921 1198651) 119889 (1198923 1198653) 119889 (1198921 1198653) + 119889 (1198923 1198651)4 = 1 0 0 24= 1

(69)

Case 3 (take points 0 and 3)

119889 (1198650 1198653) = 119889 (0 0) = 0119872 (119865 119892 0 3) = max119889 (1198920 1198923) 119889 (1198920 1198650) 119889 (1198923 1198653) 119889 (1198920 1198653) + 119889 (1198923 1198650)4 = 9 9 0 94= 9

(70)

Thus in all cases F and 119892 satisfy all the assumptions ofTheorem 23 and thus the pair F and 119892 has a unique commonfixed point 119909 = 13 An Application to an Integral Equation

This section deals with the applications of results proven inthe previous section Here we will investigate the solution ofintegral equation through our results

Consider the following integral equation

119909 (119905) = int10119870 (119905 119903 119909 (119903)) 119889119903 (71)

where119870 [0 1] times [0 1] timesR rarr R

Let 119883 = 119862[0 1] be the set of continuous real functionsdefined on [0 1] Define the b-metric by 119889(119909(119905) 119910(119905)) =max119905isin[01](|119909(119905)| + |119910(119905)|)119901 for all 119909 119910 isin 119883

Consider 119901 gt 1 Then (119883 119889) is a complete b-metric spacewith the constant 119904 = 2119901minus1119865(119909(119905)) = int1

0119870(119905 119903 119909(119903))119889119903 for all 119909 isin 119883 and for all 119905 isin[0 1] Then the existence of a solution to (71) is equivalent to

the existence of a fixed point of F Nowwe prove the followingresult

Theorem 28 Let us suppose that the following hypotheseshold

(i) 119870 [0 1] times [0 1] timesR rarr R is continuous

(ii) For all 119905 119903 isin [0 1] there exists a continuous operator120585 [0 1] times [0 1] rarr R such that

|119870 (119905 119903 119909 (119903))| + 1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816lt 1205821119901120585 (119905 119903) (|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)

sup119905isin[01]

int10120585 (119905 119903) 119889119903 le 1

(72)

where 0 lt 120582 lt 1119904(iii) There exists a function 120583 R2 rarr R such that for all119905 isin 119868 and for all 119886 119887 isin R with 120583(119886 119887) ge 0 we have

120583(1199091 (119905) int10119870 (119905 119903 119909 (119903)) 119889119903) ge 0

for 119909 isin 119862 (119868) forall119905 isin 119868120583 (119909 (119905) 119910 (119905)) ge 0

120583 (int10119870 (119905 119903 119909 (119903)) int1

0119870(119905 119903 119910 (119903)) 119889119903) ge 0

forall119905 isin 119868 forall119909 119910 isin 119862 (119868)

(73)

Then the integral equation (71) has a unique solution 119909 isin119883Proof We define 120572 119862(119868) times 119862(119868) rarr [0infin) by

120572 (119909 119910) = 1 when 120583 (119909 (119905) 119910 (119905)) ge 00 otherwise

(74)

Then for all 119909 119910 isin 119862(119868) we have 120572(119909 119910) = 1 and120572(119910 119911) = 1 implying that 120572(119909 119911) = 1 for all 119909 119910 119911 isin 119862(119868)which proves that 120572 is a transitive mapping

If 120572(119909 119910) = 1 for all 119909 119910 isin 119862(119868) then 120583(119909(119905) 119910(119905)) ge 0From (iii) we have 120583(119865119909(119905) 119865119910(119905)) ge 0 and so 120572(119865119909 119865119910) = 1Thus F is 120572-admissible

From (iii) there exists 1199091 isin 119862(119868) such that 120572(1199091 1198651199091) = 1

Abstract and Applied Analysis 9

Also we observe from (72) that

(|119865119909 (119905)| + 1003816100381610038161003816119865119910 (119905)1003816100381610038161003816)119901= (100381610038161003816100381610038161003816100381610038161003816int

1

0119870 (119905 119903 119909 (119903)) 119889119903100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816int1

0119870(119905 119903 119910 (119903)) 119889119903100381610038161003816100381610038161003816100381610038161003816)

119901

le (int10|119870 (119905 119903 119909 (119903))| 119889119903 + int1

0

1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816 119889119903)119901

le (int10(|119870 (119905 119903 119909 (119903))| + 1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816) 119889119903)

119901

le (int10(1205821119901120585 (119905 119903) (|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)) 119889119903)

119901

= (int10(1205821119901120585 (119905 119903) ((|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)119901)1119901)119889119903)

119901

le (int101205821119901120585 (119905 119903) 1198891119901 (119909 (119903) 119910 (119903)) 119889119903)119901

= 120582119889 (119909 (119905) 119910 (119905)) (int10120585 (119905 119903) 119889119903)119901

le 120582119889 (119909 (119905) 119910 (119905)) le 120582119872(119909 (119905) 119910 (119905))

(75)

which in turn implies that [120572(119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119909119910)Now all the conditions of Corollary 24 hold and F has a

unique fixed point 119909 isin 119883 which means that 119909 is the uniquesolution for the integral equation (71)

4 Conclusion

To conclude we can assert that our results are novelinteresting and generalized while considering the alpha-admissible Geraghty type mappings These results extendimprove unify and generalize many theorems based onalpha-admissible mappings in 119887-metric spaces Examples arepresented in a simplest form and illustrate the theoremsApplication to integral equation adds value to our research

Conflicts of Interest

All the authors of this article declare that they have noconflicts of interest regarding the publication of this article

Authorsrsquo Contributions

All authors have equal contribution in writing this paper Allauthors read and approved the final paper

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] R Allahyari R Arab and A S Haghighi ldquoFixed pointsof admissible almost contractive type mappings on b-metric

spaces with an application to quadratic integral equationsrdquoJournal of Inequalities and Applications vol 2015 no 32 18pages 2015

[3] M-F Bota E Karapinar andOMlesnite ldquoUlam-hyers stabilityresults for fixed point problems via 120572-120595-contractive mappingin (b)-metric spacerdquo Abstract and Applied Analysis vol 2013Article ID 825293 6 pages 2013

[4] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacerdquo Functional Analysis vol 30 pp 26ndash37 1989

[5] C Chen J Dong and C Zhu ldquoSome fixed point theorems inb-metric-like spacesrdquo Fixed Point Theory and Applications vol2015 no 122 10 pages 2015

[6] H-SDingM Imdad S Radenovic and J Vujakovic ldquoOn somefixed point results in b-metric rectangular and b-rectangularmetric spacesrdquo Arab Journal of Mathematical Sciences vol 22(2016) pp 151ndash164 2015

[7] S Radenovic M Jovanovic and Z Kadelburg ldquoCommon fixedpoint results in metric-type spacesrdquo Fixed Point Theory andApplications vol 2010 article 978121 15 pages 2010

[8] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 article 315398 2010

[9] J R Roshan V Parvaneh and Z Kadelburg ldquoCommon fixedpoint theorems for weakly isotone increasing mappings inordered b-metric spacesrdquo Journal of Nonlinear Science and itsApplications JNSA vol 7 no 4 pp 229ndash245 2014

[10] W Sintunavarat S Plubtieng and P Katchang ldquoFixed pointresult and applications on a b-metric space endowed with anarbitrary binary relationrdquo Fixed Point Theory and Applicationsvol 2013 article 296 2013

[11] W Sintunavarat ldquoGeneralized Ulam-Hyers stability well-posedness and limit shadowing of fixed point problems for 120572-120573-contraction mapping in metric spacesrdquo The Scientific WorldJournal vol 2014 Article ID 569174 7 pages 2014

[12] R J Shahkoohi and A Razani ldquoSome fixed point theoremsfor rational Geraghty contractivemappings in ordered b-metricspacesrdquo Journal of Inequalities and Applications vol 373 p 232014

[13] S L Singh and B Prasad ldquoSome coincidence theorems andstability of iterative proceduresrdquoComputersampMathematics withApplications vol 55 no 11 pp 2512ndash2520 2008

[14] M A Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[15] S Cho J Bae and E Karapınar ldquoFixed point theorems for 120572-Geraghty contraction type maps in metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 329 11 pages 2013

[16] D Dukic Z Kadelburg and S Radenovic ldquoFixed points ofGeraghty-type mappings in various generalized metric spacesrdquoAbstract and Applied Analysis vol 2011 Article ID 561245 13pages 2011

[17] M E Gordji M Ramezani Y J Cho and S Pirbavafa ldquoAgeneralization ofGeraghtyrsquos theorem in partially orderedmetricspaces and applications to ordinary differential equationsrdquoFixed Point Theory and Applications vol 2012 article 74 2012

[18] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

10 Abstract and Applied Analysis

[19] A Felhi S Sahmim and H Aydi ldquoUlam-hyers stability andwell-posedness of fixed point problems for 120572-120582-metric spacesrdquoFixed Point Theory and Applications vol 2016 no 1 20 pages2016

[20] V L Rosa and P Vetro ldquoCommon fixed points for 120572-120595-120601-contractions in generalized metric spacesrdquo Nonlinear AnalysisModelling and Control vol 19 no 1 pp 43ndash54 2014

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

Also we observe from (72) that

(|119865119909 (119905)| + 1003816100381610038161003816119865119910 (119905)1003816100381610038161003816)119901= (100381610038161003816100381610038161003816100381610038161003816int

1

0119870 (119905 119903 119909 (119903)) 119889119903100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816int1

0119870(119905 119903 119910 (119903)) 119889119903100381610038161003816100381610038161003816100381610038161003816)

119901

le (int10|119870 (119905 119903 119909 (119903))| 119889119903 + int1

0

1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816 119889119903)119901

le (int10(|119870 (119905 119903 119909 (119903))| + 1003816100381610038161003816119870 (119905 119903 119910 (119903))1003816100381610038161003816) 119889119903)

119901

le (int10(1205821119901120585 (119905 119903) (|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)) 119889119903)

119901

= (int10(1205821119901120585 (119905 119903) ((|119909 (119903)| + 1003816100381610038161003816119910 (119903)1003816100381610038161003816)119901)1119901)119889119903)

119901

le (int101205821119901120585 (119905 119903) 1198891119901 (119909 (119903) 119910 (119903)) 119889119903)119901

= 120582119889 (119909 (119905) 119910 (119905)) (int10120585 (119905 119903) 119889119903)119901

le 120582119889 (119909 (119905) 119910 (119905)) le 120582119872(119909 (119905) 119910 (119905))

(75)

which in turn implies that [120572(119909 119910) minus 1 + 120575]119889(119865119909119865119910) le 120575120582119872(119909119910)Now all the conditions of Corollary 24 hold and F has a

unique fixed point 119909 isin 119883 which means that 119909 is the uniquesolution for the integral equation (71)

4 Conclusion

To conclude we can assert that our results are novelinteresting and generalized while considering the alpha-admissible Geraghty type mappings These results extendimprove unify and generalize many theorems based onalpha-admissible mappings in 119887-metric spaces Examples arepresented in a simplest form and illustrate the theoremsApplication to integral equation adds value to our research

Conflicts of Interest

All the authors of this article declare that they have noconflicts of interest regarding the publication of this article

Authorsrsquo Contributions

All authors have equal contribution in writing this paper Allauthors read and approved the final paper

References

[1] S Czerwik ldquoContraction mappings in b-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[2] R Allahyari R Arab and A S Haghighi ldquoFixed pointsof admissible almost contractive type mappings on b-metric

spaces with an application to quadratic integral equationsrdquoJournal of Inequalities and Applications vol 2015 no 32 18pages 2015

[3] M-F Bota E Karapinar andOMlesnite ldquoUlam-hyers stabilityresults for fixed point problems via 120572-120595-contractive mappingin (b)-metric spacerdquo Abstract and Applied Analysis vol 2013Article ID 825293 6 pages 2013

[4] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacerdquo Functional Analysis vol 30 pp 26ndash37 1989

[5] C Chen J Dong and C Zhu ldquoSome fixed point theorems inb-metric-like spacesrdquo Fixed Point Theory and Applications vol2015 no 122 10 pages 2015

[6] H-SDingM Imdad S Radenovic and J Vujakovic ldquoOn somefixed point results in b-metric rectangular and b-rectangularmetric spacesrdquo Arab Journal of Mathematical Sciences vol 22(2016) pp 151ndash164 2015

[7] S Radenovic M Jovanovic and Z Kadelburg ldquoCommon fixedpoint results in metric-type spacesrdquo Fixed Point Theory andApplications vol 2010 article 978121 15 pages 2010

[8] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications vol 2010 article 315398 2010

[9] J R Roshan V Parvaneh and Z Kadelburg ldquoCommon fixedpoint theorems for weakly isotone increasing mappings inordered b-metric spacesrdquo Journal of Nonlinear Science and itsApplications JNSA vol 7 no 4 pp 229ndash245 2014

[10] W Sintunavarat S Plubtieng and P Katchang ldquoFixed pointresult and applications on a b-metric space endowed with anarbitrary binary relationrdquo Fixed Point Theory and Applicationsvol 2013 article 296 2013

[11] W Sintunavarat ldquoGeneralized Ulam-Hyers stability well-posedness and limit shadowing of fixed point problems for 120572-120573-contraction mapping in metric spacesrdquo The Scientific WorldJournal vol 2014 Article ID 569174 7 pages 2014

[12] R J Shahkoohi and A Razani ldquoSome fixed point theoremsfor rational Geraghty contractivemappings in ordered b-metricspacesrdquo Journal of Inequalities and Applications vol 373 p 232014

[13] S L Singh and B Prasad ldquoSome coincidence theorems andstability of iterative proceduresrdquoComputersampMathematics withApplications vol 55 no 11 pp 2512ndash2520 2008

[14] M A Geraghty ldquoOn contractive mappingsrdquo Proceedings of theAmerican Mathematical Society vol 40 pp 604ndash608 1973

[15] S Cho J Bae and E Karapınar ldquoFixed point theorems for 120572-Geraghty contraction type maps in metric spacesrdquo Fixed PointTheory and Applications vol 2013 no 329 11 pages 2013

[16] D Dukic Z Kadelburg and S Radenovic ldquoFixed points ofGeraghty-type mappings in various generalized metric spacesrdquoAbstract and Applied Analysis vol 2011 Article ID 561245 13pages 2011

[17] M E Gordji M Ramezani Y J Cho and S Pirbavafa ldquoAgeneralization ofGeraghtyrsquos theorem in partially orderedmetricspaces and applications to ordinary differential equationsrdquoFixed Point Theory and Applications vol 2012 article 74 2012

[18] B Samet C Vetro and P Vetro ldquoFixed point theorems for120572-120595-contractive type mappingsrdquo Nonlinear Analysis TheoryMethods and Applications vol 75 no 4 pp 2154ndash2165 2012

10 Abstract and Applied Analysis

[19] A Felhi S Sahmim and H Aydi ldquoUlam-hyers stability andwell-posedness of fixed point problems for 120572-120582-metric spacesrdquoFixed Point Theory and Applications vol 2016 no 1 20 pages2016

[20] V L Rosa and P Vetro ldquoCommon fixed points for 120572-120595-120601-contractions in generalized metric spacesrdquo Nonlinear AnalysisModelling and Control vol 19 no 1 pp 43ndash54 2014

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

[19] A Felhi S Sahmim and H Aydi ldquoUlam-hyers stability andwell-posedness of fixed point problems for 120572-120582-metric spacesrdquoFixed Point Theory and Applications vol 2016 no 1 20 pages2016

[20] V L Rosa and P Vetro ldquoCommon fixed points for 120572-120595-120601-contractions in generalized metric spacesrdquo Nonlinear AnalysisModelling and Control vol 19 no 1 pp 43ndash54 2014

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of